Calculation of the photoemission spectrum of
fullerene
Gattari Paolo
July, 2003
advisor: Nicola Manini
http://www.mi.infm.it/manini/theses/gattari.pdf
ii
Contents
Abstract
v
1 Photoemission spectroscopy
1.1 Introduction . . . . . . . . . . . . . . .
1.2 Basic features of photoelectron spectra
1.2.1 Atoms . . . . . . . . . . . . . .
1.2.2 Molecules . . . . . . . . . . . .
1.3 Fullerene C60 . . . . . . . . . . . . . .
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2 Molecular physics
2.1 The Born-Oppenheimer approximation . .
2.2 Electronic states . . . . . . . . . . . . . .
2.3 Molecular vibrations . . . . . . . . . . . .
2.4 The full molecular Hamiltonian . . . . . .
2.5 Symmetry and vibronic states classification
2.6 Dynamic Jahn-Teller effect in molecules . .
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3 The
3.1
3.2
3.3
3.4
3.5
model for dynamic Jahn-Teller in C60
The fullerene molecule . . . . . . . . . . .
The electronic structure . . . . . . . . . .
The vibrational modes . . . . . . . . . . .
e-v coupling: symmetry considerations . .
The hu ⊗ (Hg + Gg + Ag ) model . . . . . .
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4 The
4.1
4.2
4.3
photoemission spectrum
Fermi’s Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . .
Green’s function approach . . . . . . . . . . . . . . . . . . . . . . .
Photoemission: the sudden approximation . . . . . . . . . . . . . .
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5 Nondegenerate vibrational modes
5.1 x ⊗ A - shifted oscillator . . . . . . . . . . . . . . . . . . . . . . . .
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iv
CONTENTS
5.2 The A modes in a many-mode context . . . . . . . . . . . . . . . .
5.3 Two A modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 66 A modes: a fictitious spectrum of C60 . . . . . . . . . . . . . . .
6 Degenerate modes
6.1 The Lanczos algorithm . . . . . . . . .
6.2 The JT modes of C60 : Lanczos spectra
6.3 PES of degenerate modes . . . . . . . .
6.4 Finite-temperature Lanczos . . . . . .
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7 Analysis and discussion
7.1 T = 0 analysis . . . . . . . . . . . . . . . .
7.2 1-phonon vibronic multiplet interpretation
7.3 Thermal effects . . . . . . . . . . . . . . .
7.4 Conclusions and outlook . . . . . . . . . .
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Abstract
Photoemission from a degenerate shell of a high-symmetry molecule, as the outer
shell of C60 , is necessarily accompanied by characteristic vibronic structures in the
measured spectrum, associated to Jahn Teller (JT) splitting of the orbital degeneracy through electron-phonon interaction in the final state. The experimental C60
spectrum [1, 2] shows a characteristic strong broad satellite at 230 meV from the
main peak, significantly higher in energy than any characteristic vibrational frequency of C60 .
The purpose of the present work is to understand and explain quantitatively the
measured spectrum. We use a linear and harmonic Jahn-Teller model, based on the
DFT-LDA frequencies and couplings of Ref. [3] (including all the 66 active modes),
to compute numerically and virtually exactly the C60 photoemission spectrum from
the HOMO (highest occupied molecular orbital). The computed lineshape shows
dominant features close to experiment, in particular the satellite at 230 meV (left
panel),
hu
α=0.5
t1u
gu
au
(7)
0.5
average
(8)
Intensity [arb. units]
computed spectrum (T=0 K)
1
& Hg
(Energy-EGS) / hω
t2u
Hg1 spectrum (T=800K)
Hg
experiment
hu (GS)
0
-200
0
200
400
600
0
2
4
6
Jahn-Teller coupling g
Binding Energy [meV]
Our analysis explains this satellite in terms of the JT-vibronic splitting of the
highest frequency Hg modes (Hg 7, 180 meV and Hg 8, 197 meV). The increase in
v
vi
ABSTRACT
vibron energy is a characteristic signature of the dynamic JT effect and can be
understood on the basis of symmetry: the symmetry hu of the bare hole produced
by photoemission from the HOMO, remains hu even in the presence of interacting
vibrations. On the other hand, the average of the multiplet of states derived from
the 1-vibration manifold is, to second order in the electron-phonon coupling g,
independent of g, because the JT coupling is a traceless perturbation within that
manifold. Within this multiplet (made of Hg × hu = au + t1u + t2u + 2gu + 2hu ), the
hu vibronic states are the sole that have another hu state (the GS itself) lower in
energy and “pushing” them upward. (right panel). Our calculations confirm that
the DFT-LDA parameters of Ref. [3] account for this energy increase, from 190 to
230 meV.
Technically, to compute the spectrum we separate the degenerate modes from the
nondegenerate ones. We compute the contribution of the quite medium JT-coupled
degenerate Hg and Gg modes, by Lanczos diagonalization of the electron-vibration
Hamiltonian by continued fraction expansion of the finite-temperature spectrum,
assuming thermal equilibrium of the sample. The trivial nondegenerate Ag modes
distort the molecule without splitting the orbital degeneracy. Their contribution is
evaluated analytically and their effect is a weak general broadening of the spectrum.
However, the T = 0 spectrum appears much more resolved than the experimental
one, which is measured at 800 K.
The technique used, very efficient at T = 0, gives converging spectra at low temperature (< 100 K), but it becomes useless at higher temperature, due to the large
number of initial thermal excitations. However, thermal effects influence mostly the
lowest-frequency Hg -mode (which is the most strongly coupled). The inclusion of
the Hg 1 mode only, for which our technique gives converging results up to 800 K,
is sufficient to account for most of the experimental broadening (left panel).
Chapter 1
Photoemission spectroscopy
1.1
Introduction
When light of short enough wavelength interacts with free molecules, it can cause
electrons to be ejected from the occupied molecular orbitals. Photoelectron spectroscopy [4] is the study of the photoelectrons, whose energy, abundances and angular distribution are all characteristic of the individual molecular orbitals from which
they originate. The quantity measured most directly in photoemission spectroscopy
is the ionization potential for the removal of electrons from different molecular orbitals. It is a well known basic result (Koopmans’ theorem) that each ionization
potential Ij is equal in magnitude to an orbital energy, j :
Ij = −j
(1.1)
This approximate results is very useful in that it indicates that the photoelectron
spectrum of a molecule is a direct representation of the molecular orbital energies.
Moreover, for a molecule, the photoemission spectrum gives not only the orbital
energies but also, less directly, the changes in molecular geometry caused by the
removal of one electron from each orbital. These change reveal the character of the
orbitals, whether they are bonding, antibonding or non-bonding, and where their
bonding power is localized in the molecule.
1.2
Basic features of photoelectron spectra
In a photoelectron spectrometer, an intense beam of monochromatic ultraviolet or
X-ray light ionizes the molecules or atoms of a gas in a ionization chamber:
M + h̄ω → M + + e
1
(1.2)
2
CHAPTER 1. PHOTOEMISSION SPECTROSCOPY
Figure 1.1: Idealized photoionizzation process and photoelectron spectrum of an atom
The light used must have an energy sufficient to ionize electrons at least from the
highest valence shell of molecules or atoms, that is the highest occupied molecular
orbital (HOMO). If h̄ω is larger, electrons may be ejected also from deeper levels.
In each orbital j of an atom or molecule, the electrons have a characteristic binding
energy, the minimum energy needed to eject them to infinity. Part of the energy
of the photon is used to overcome this binding energy, |j |, and, if the species is an
atom, the remainder, h̄ω − |j |, must appear as kinetic energy (Eke ) of the ejected
electron:
Eke = h̄ω − |j |
(1.3)
The ejected photoelectron are separated according to their kinetic energy in an
electron energy analyzer, detected and recorded. The photoelectron spectrum is a
record of the number of electrons detected at each energy: a peak is found in the
spectrum at each energy h̄ω − Eke corresponding to the binding energy |j | of an
electron in the atom, as illustrated schematically in Fig. 1.1.
If the species is a molecule, there are the additional possibilities of vibrational
1.2. BASIC FEATURES OF PHOTOELECTRON SPECTRA
3
and/or rotational excitation upon ionization, so the energy of the photoelectrons
may be reduced:
ion
|j | − Evib.,rot.
= h̄ω − Eke
(1.4)
The spectrum may now be decorated by several vibrational structure for each orbital: the system of lines that correspond to ionization from a single molecular
orbital constitutes a band.
Apart from Koopmans’ theorem, there are two simple rules that usually make
the relationship between photoelectron spectra and molecular electronic structure
especially simple: 1) Each band in the spectrum corresponds to ionization from a
single molecular orbital.
2) Each occupied molecular orbital of binding energy less than h̄ω gives rise to a
single band in the spectrum.
Because of these rules, the photoelectron spectrum is a simple reflection of the
molecular orbital diagram (Fig 1.1). These rules are a simplification, however,
and there are five reasons why there may, in fact, be more or fewer bands in a
spectrum than there are valence orbitals in a molecule. Firstly, additional bands are
sometimes found that correspond to the ionization of one electron with simultaneous
excitation of a second electron to an unoccupied excited orbital. This is a twoelectron process, and the bands produced in the spectrum are normally much weaker
than simple ionization bands. Secondly, ionization from a degenerate occupied
molecular orbital can give rise to as many bands in the spectrum as there are orbital
components, because although the orbitals are degenerate in the molecule, they may
not be so in the positive ion. The mechanisms that remove the degeneracy are spinorbit coupling and Jahn-Teller effect. Thirdly, ionization from molecules like O2
or NO, which have unpaired electrons, can give many more bands than there are
occupied orbitals in the molecules, and in such instances neither Koopmans’ theorem
nor the simple rules apply. Fourthly, molecular orbitals lying close in energy may
give rise to overlapping bands. Finally, finite temperature and the finite life time of
excited electronic states usually lead to broadening and thus to the obliteration of
some spectral features. In order to introduce these main features of photoelectron
spectra, it is convenient to take practical examples, starting with the spectra of
single atoms and proceeding to those of more complicated molecules.
1.2.1
Atoms
A typical atomic photoelectron spectrum is shown in Fig. 1.2 This is the photoemission spectrum of atomic mercury excited by helium resonance radiation. The
vertical scale is the strength of the electron signal, usually given in electron per
second. The absolute intensities have no physical significance because they depend on physical and experimental factors which, although constant throughout
4
CHAPTER 1. PHOTOEMISSION SPECTROSCOPY
Figure 1.2: Photoemission spectrum of mercury excited by He I radiation (from Ref. [4]).
the measurement of the spectrum, are not precisely known. The relative intensities
of different peaks in the spectrum are meaningful, however, as they are equal to
the relative probability of photoionizzation to different states of the positive ion,
which are called the relative partial ionization cross-sections. Three horizontal scale
are given on Fig. 1.2 to illustrate the relationship between measured electron energy, ionization potential and the internal excitation energy of the ions, including
electronic excitation energy.
The spectrum in Fig. 1.2 show that Hg+ ions are formed by photoionization
in three electronic states, with ionization energies of 10.44, 14.84, and 16.71 eV.
The states involved are well known from the atomic spectrum of mercury and have
the designation 2 S 1 , 2 D 5 and 2 D 3 , respectively. In particular, the 2 S 1 state of
2
2
2
2
Hg+ is produced by the ejection of one of the 6s electrons (the neutral electron
configuration is 5d10 6s2 ), but both the 2 D states are produced by the ejection of 5d
electrons. Notice the energy difference between the 2 D 5 and 2 D 3 states arising from
2
2
5d ionization: it represents a breakdown of the rule of one band per orbital, in this
instance due to spin-orbit coupling. Then the Koopmans’ theorem (1.1) cannot be
used directly to derive the orbital energy for 5d electron in mercury, and a weighted
mean of all the energies for ionization for a single orbitals must be taken, where the
weights are the statistical weight of the ionic states produced.
In spite of these small intricacies the photoemission spectrum of an atom is quite
1.2. BASIC FEATURES OF PHOTOELECTRON SPECTRA
5
Figure 1.3: Photoelectron spectrum of nitrogen N 2 excited by He I radiation (from
Ref. [4]).
clean and the characteristic structures can be relatively easily interpreted.
1.2.2
Molecules
The photoemission spectrum of the diatomic molecule N2 is shown in Fig. 1.3 as
the next step in the hierarchy of complication. Three electronic states of N+
2 are
reached by photoionization, and they appear in the spectrum as the sharp peak at
15.6 eV, the group of peaks between 16.7 and 18 eV and the weak peak at 18.8 eV.
Each electronic state actually gives a group of peaks in the spectrum because of
the possibility of vibrational as well as electronic excitation. Every resolved peak
in the spectrum of a molecule is a single vibrational line and represents a definite
number of quanta of vibrational energy in the molecular ion.
As was the case for mercury, the ionic states of N+
2 seen in the photoemission
spectrum are well characterized by other forms of spectroscopy. They have the
2
2 +
designation X 2 Σ+
g , A Πu , and B Σu in order of increasing ionization potential,
−1
and correspond to the process σg , πu−1 and σu−1 respectively.
It is clear that the bands in the spectrum of this three ionization are very different
both in the spacing of the lines within each bands, i.e. the sizes of the vibrational
quanta in the ions, and also in the intensity of the vibrational lines. The spacings
6
CHAPTER 1. PHOTOEMISSION SPECTROSCOPY
Figure 1.4: Photoelectron spectrum of oxygen O 2 excited by He I radiation (from
Ref. [4]).
between the lines depends on the vibrational frequencies in the different electronic
ion
:
states of the ion, since for the vibrational excitation energies Evib
1
ion
Evib
= (v + )h̄ω
2
(1.5)
Here v is the vibrational quantum number and ω the vibrational frequency, which
depends on the strength of the N-N bond in the different electronic states, Therefore
if a bonding electron is removed, the bond becomes weaker, and the vibrational
frequency ω becomes lower than in the neutral molecule. This is exactly what
happens in πu−1 ionization of N2 , where the frequency drops from 2360 cm−1 in the
molecule to 1800 cm −1 in the A 2 Πu state of N+
2 , indicating the strongly bonding
character of πu electrons. Vice versa any antibonding character of the electrons
removed on ionization is revealed by an increase in frequency, and an example of
this is the πg−1 ionization of molecular oxygen (giving a single state in the molecular
ion, X2 Πg and a single band in the spectrum), shown in Fig. 1.4.
The relative intensities of the vibrational lines in a ionization band are also
related to the bonding power of the removed electron. Strong vibrational excitation
is associated to a change in equilibrium bond length up on ionization and the relation
between them is given by the Franck-Condom principle. According to this principle
when strongly bonding electrons are removed many peaks appear corresponding to
different final vibrational states of the ion. For antibonding electrons instead one, or
1.2. BASIC FEATURES OF PHOTOELECTRON SPECTRA
7
Figure 1.5: Photoelectron spectrum of water H 2 O excited by He I light (from Ref. [4]).
few peaks appear, given by the transition to the ground or lower excited vibrational
states.
The spectra of molecules with more than two atoms are naturally more complex,
because there are generally more molecular orbitals from which ionization can take
place and many different modes of vibration that may be excited by ionization. As
example, in Fig. 1.5 we show the photoemission spectrum of the water. H2 O is a
simple triatomic molecule, with 3 modes of vibration, and its spectrum is still quite
clean.
One important principle is that the vibration that corresponds most closely to
the change in equilibrium molecular geometry caused by a particular ionization
will be one that is most strongly excited. In a band that shows excitation of
several modes, it can easily happen that the vibrational structure is so complex
that the individual lines cannot be resolved and a continuous contour is observed.
Such unresolved bands are, in fact, much more common that resolved bands in the
spectra of molecules with five or more atoms.
In addition to the complexity of overlapping vibrational structures, one more
reason of the presence of continuous bands in the spectra is short lifetime of the ion
in the molecular ionic states in which they are initially formed. If a molecular ion in
a given state has a lifetime τ for dissociation, radiative decay or internal conversion
to another electronic state, the original state will have an energy width ∆E, given
8
CHAPTER 1. PHOTOEMISSION SPECTROSCOPY
Figure 1.6: The gas phase photoemission spectrum is a probe of the valence orbitals of
C60 , while matrix-isolated C 1s XAS data probe its empty molecular orbitals. The data
are placed on a common binding energy scale (from Ref. [5]).
by the uncertainty principle:
h̄
.
(1.6)
τ
This energy uncertainty causes a broadening of all spectral lines and may make the
band continuous. This cause of broadening occurs even in the photoelectron spectra
of diatomic molecules.
∆E ∼
1.3
Fullerene C60
Figure 1.6 shows a wide-range gas-phase photoemission spectrum of fullerene C60 ,
the main object of our investigation. One can immediately get a sense of the high
degree of orbital degeneracy which is the hallmark of C60 : although the system has
60 π-electrons, the photoemission spectrum shows only an handful of distinct bands
at low binding energy. It is in fact relatively straightforward to correlate the features
in the spectrum with the different molecular orbitals of icosahedral C60 . The peaks
nearest the chemical potential originates from the hu HOMO. The peak which comes
next in energy is due to the hg and gg states and is generally referred as the HOMO-1.
Both the HOMO and HOMO-1 are pure π molecular orbitals. The features at higher
1.3. FULLERENE C60
9
energy results from the overlaps of both π and σ molecular orbitals. Figure 1.6 shows
also the corresponding C 1s excitation spectrum of matrix isolated C60 measured
using XAS, and placed on a common binding energy scale, revealing the t1u LUMO
(Lowest Unoccupied Molecular Orbitals), as well as structures at higher energy.
In this work we are interested to investigate the vibronic structures that necessarily accompany the photoemission from the HOMO. In fact photoemission from
a closed-shell high-symmetry molecule, as C60 , involves the Jahn-Teller effect in the
final states, splitting the orbitals degeneracy, and this reflects in a higher complexity
of the corresponding band. We therefore concentrate in understanding the detailed
features of the broad peak labeled HOMO in Fig. 1.6.
10
CHAPTER 1. PHOTOEMISSION SPECTROSCOPY
Chapter 2
Molecular physics
The purpose of this chapter is to summarize the basic theoretical concepts of molecular physics and single out the approximations which lead to the choice of the model,
described in detail in the following chapter, for photoemission of a molecule such
as C60 .
If the wave equation for a molecule composed of electrons and nuclei is set up,
a procedure (the so-called Born-Oppenheimer approximation) exists whereby this
equation may be separated into two equations, one of which governs the electronic
motions and yields the forces holding the atoms together, whereas the other is
the equation for motions of rotation and vibration of the positive ions. On the
basis of this approximation, the forces between the atoms are routinely calculated
numerically from the electronic wave equation, by means of standard techniques
generically indicated by ab-initio methods. The separation of the electronic and
ionic motions is an approximation which often works very well, but can break down
in the presence of electronic degeneracies or large excitation energy of the ionic
motion [6].
2.1
The Born-Oppenheimer approximation
The total non relativistic Hamiltonian for a molecule can be written
2
X Zk Zt e 2
X e2
h̄2 X ∇R~ k
h̄2 X 2 X Zk e2
H=−
−
∇~ri −
+
+
~
~k − R
~ t|
2 k Mk
2me i
|~ri − ~rj |
ri | k>t |R
i>j
k,i |Rk − ~
(2.1)
where i, j refer to electrons and k, t refer to nuclei, Mk is the mass of the k-th nucleus,
qe2
is the electromagnetic coupling constant,
me is the mass of the electron, e2 = 4π
0
with qe representing the elementary charge. Indicating collectively with R all the
~ k , and r all the electron coordinates ~ri , the Hamiltonian (2.1)
nuclear coordinates R
11
12
CHAPTER 2. MOLECULAR PHYSICS
can be written more compactly as
H = Tn + Te + Ven (r, R) + Vnn (R) + Vee (r)
(2.2)
where Tn and Te are the nuclear and electronic kinetic energies, Ven is the electronion attraction, Vnn and Vee are the ion-ion and the electron-electron repulsion respectively. The Ven term prevents us from separating H into nuclear and electronic
parts, thus separating the molecular wavefunction into a product of nuclear and
electronic terms
Ψ(r, R) = ψ(r)φ(R).
(2.3)
Moreover, the attractive term Ven is large and can neither be neglected nor treated
as a perturbation. It is responsible, in particular, of the cusp-like behavior of the
~ k , and, more importantly, of chemical bonding.
electronic wavefunction at ~ri → R
The best progress toward a reasonable separation of the nuclear and electronic
motion can be achieved by introducing a parametric R dependence in the electronic
wavefunction, so that the total wavefunction reads
Ψ(r, R) = ψ(r; R)φ(R).
(2.4)
This ansatz is justified by the following observations: a) The electron mass m
is much smaller than the ionic mass Mk , thus the time scale for the electronic
motion is much faster than that for the ionic movement; b) Different energy levels
corresponding to different electronic eigenstates are separated by energy gaps which
are large with respect to the corresponding separation between ionic eigenstates (i.e.
with respect to the characteristic energies associated to the motion of the ions).
According to this adiabatic (or Born-Oppenheimer) approximation, once an electronic state has been selected, the ions move without inducing transitions between
different electronic states. The electronic wavefunction ψ(r; R) hence describes the
electronic state corresponding to a given frozen geometrical configuration of the
nuclei. By definition ψ(r; R) satisfies the parametric Schrödinger equation:
He ψα (r; R) = Uα (R)ψα (r; R),
(2.5)
He = Te + Ven (r, R) + Vnn (R) + Vee (r)
(2.6)
where
With α we represent the set of quantum numbers characterizing a given eigenstate
with electronic energy Uα (R), to be recognized as the (adiabatic) potential energy
surface that governs the motion of the nuclei. Notice that ψα (r; R) is a many-body
eigenstate, since no mean field approximation has been introduced yet.
Consider again the original complete Hamiltonian (2.2). It can be rewritten
using (2.6)
H = Tn + He
(2.7)
2.1. THE BORN-OPPENHEIMER APPROXIMATION
13
The most general solution of the complete molecular Schrödinger equation
(H − E)Ψ(r, R) = 0
(2.8)
can be obtained, in terms of the individual electronic wavefunctions ψα (r; R), by
using an (in principle infinite) expansion of the form
X
Ψ(r, R) =
ψα (r; R)φα (R).
(2.9)
α
Here φ(R) are nuclear-coordinate dependent coefficients, we interpret as the nuclear wave functions. To the extent that the Born-Oppenheimer approximation is
valid, accurate solution can be obtained using only one (Eq. (2.4)) or a few terms.
Substituting Eq. (2.9) into Eq. (2.8), multiplying it from the left by ψβ∗ (r; R) and integrating over the electronic coordinates while recalling Eqs. (2.7) and (2.5), yields
the following set of coupled equations
Z
ψβ∗ (r; R)Tn φα (R)ψα (r; R)dr + Uβ (R) − E φβ (R) = 0
(2.10)
where, from now to the end of this section, we omit the sum over the repeated
index α. Using the explicit expression for Tn the molecular kinetic term gives three
contributions: one adiabatic term involving only the nuclear wavefunction
Tn φβ (R) =
X
k
−
h̄2 2
∇ φβ (R)
2Mk R~ k
(2.11)
and two non-adiabatic contributions coupling the electronic and nuclear motion
X h̄ Z 1
Nβα φα (R) =
−
ψβ (r; R)∇R~ k ψα (r; R) dr ∇R~ k φα (R)
(2.12)
Mk
k
and
2
Nβα
φα (R)
=
X
k
h̄
−
2Mk
Z
ψβ (r; R)∇2R~ ψα (r; R) dr φα (R)
k
The Schrödinger equation is finally rewritten compactly as
1
2
Tn + Uβ (R) − E φβ (R) + Nβα
+ Nβα
φα (R) = 0
(2.13)
(2.14)
The non-adiabatic terms N 1 and N 2 , coupling together the motions associated to
different electronic eigenvalues, become negligible if the variation of ψα (r; R) with
the nuclear coordinates is sufficiently slow. Born and Oppenheimer have shown,
using the mathematical expression of the arguments at points a and b above, that
14
CHAPTER 2. MOLECULAR PHYSICS
this conditions is usually fulfilled to a satisfactory approximation. In this case the
nuclear equation is totally decoupled from the electronic ones and it can be written
Tn + Uβ (R) − Eβ,w φβ,w (R) = 0
(2.15)
With w, we represent the set of quantum number characterizing a given ionic eigenstate φβ,w (R) with eigenvalues Eβ,w where the subscript β labels the choice of the
electronic state. Uβ (R) are the adiabatic molecular potential energy surfaces that
govern the motion of the nuclei. They contain the direct Coulomb repulsive nuclear
interaction Vnn and the electronic contributions that can be considered as the “glue”
which keeps the atoms together. Uβ (R) are in general complicated functions of all
the nuclear coordinates R. In particular, they cannot be generally expressed as a
simple sum of two body contributions.
Every electronic state of a polyatomic molecule is characterized by a different
potential surface. If a given electronic potential surface Uβ (R) shows no minimum
as a function of R, that electronic state is unstable and the molecule in that state
is likely to dissociate. If Uβ (R) has at least one deep enough global minimum R0 ,
then the molecule is stable in the electronic state β. The minimum Uβ (R0 ) of the
potential surface of a given stable electronic state is considered as the electronic
energy of this state and designated Eβe .
The total energy may then be written, in this approximation
vib
Eβ,w = Eβe + Eβ,w
(2.16)
vib
is the vibrational energy obtained from the solution of the Schrödinger
where Eβ,w
equation for the ions Eq. (2.15). The corresponding adiabatic eigenstate is then
simply as in (2.4)
Ψβ,w (r, R) = ψβ (r; R)φβ,w (R)
(2.17)
(β)
The potential minima R0 of different electronic states occur, in general, at different values of internuclear distances and angles often characterized by different
geometrical symmetries. Moreover, potential surfaces with several minima occur
not infrequently (sometimes corresponding to different isomers) [7], for example for
vib
the electronic ground state of NH3 and cis/trans difluoro etylene (CHF)2 . In Eβ,w
the roto-translational contributions are tacitly included, which can be seen as zerofrequency normal vibrations essentially decoupled from the other ones. They can
be ignored if we choose a reference system rigidly tied to the molecule.
There is a whole class of interesting situations where the Born-Oppenheimer
approximation fails. The non-adiabatic terms in (2.14) become sizable and in the
expression for the total energy (2.16) non-adiabatic contributions need to be accounted for. These terms couple electronic and nuclear motion, inducing transitions
between electronic states. Typical examples are molecules with degenerate or almost degenerate ground state, where transitions between the degenerate states are
15
2.2. ELECTRONIC STATES
easily induced by this coupling. The true eigenstates then will be expanded as linear
combinations of the Born-Oppenheimer solutions (2.17)
X g
Ψg (r, R) =
cβ,w ψβ (r; R)φβ,w (R).
(2.18)
β,w
Here clβ,w are expansion coefficients, and the subscript g has been added as a reminder that there are multiple solutions to the Schrödinger equation. (2.18) is an
infinite expansion, but normally few terms are needed to get accurate solutions.
2.2
Electronic states
The equation for the electronic system (2.5) is a many-body equation depending on
the coordinates r = ~r1 , ~r2 . . . of all the electrons of the system and parametrically on
the nuclear configuration R. It must necessarily be treated by standard approximate
methods of quantum many-body theory, such as the Hartree-Fock method [8], or
the Density Functional Theory [9], or Quantum Monte Carlo techniques [10].
As mentioned in section above, in a first approximation (which is normally a
good approximation) the electronic motion is studied at the equilibrium positions
of the ions. Therefore, for the rest of this section, we do not indicate explicitly the
dependencies of the electronic eigenfunctions from the nuclear equilibrium configurations R0 .
Mean-field approximations typically reduce Eq. (2.5) to a single one-body equation for the one-electron wavefunctions of the system, typically to be solved selfconsistently. If ai and ψ̃ai (~ri ) are respectively the eigenvalues and the corresponding
eigenstates of this one-body equation, the total electronic energy can be written
Eαe = E M B + a1 + a2 + . . . + ane
(2.19)
where E M B is some many-body energy and the corresponding eigenstates are in the
form of Slater determinants of single-particle wavefunctions:
1 X
ψα (r) = √
(2.20)
(−)P ψ̃aP1 (~r1 )ψ̃aP2 (~r2 ) . . . ψ̃aPne (~rne )
ne ! P
where ne is the total number of electrons of the molecule, ai are the sets of quantum
numbers for the single particle problem and α = (a1 , . . . an ) is given in terms of
those. The summation involves all the permutations P of the ne quantum numbers.
In the second-quantized language, the electronic Hamiltonian can be represented
on the basis of the fixed single-particle orbitals as
X
He =
a c†a ca + E M B
(2.21)
a
16
CHAPTER 2. MOLECULAR PHYSICS
where the operator c†a creates an electron in the single particle level a and ca is the
corresponding annihilator. The operator Na = c†a ca gives the number of electrons
in spin-orbital a and it can be either 0 or 1 for fermions. The many-body term in
Eq. (2.21)
E M B = Fee ({c† }, {c})
(2.22)
is a function of all the creators and annihilators, globally indicated with {c† } and
{c}. The subscript ee underlines the electron-electron character of this interaction.
To treat electron correlations explicitly, nontrivial linear combinations of Slater
determinants (2.20) may be considered (configuration interaction - CI - method).
2.3
Molecular vibrations
Useful information for the solution of the adiabatic equation (2.15), describing the
motion of the ions, are obtained within a classical scheme. Specifically, the classical
small oscillations around the equilibrium positions, form the basis for the solution
of the quantum problem (2.15).
Consider the adiabatic potential energy surface Uα (R) governing the motion of
the nuclei, given by equation (2.5). The dependence on the electronic state α will be
~i − R
~ 0 i to represent the displacement
left implicit in this Section. By taking ~ui = R
~ 0 i , we can write an expansion
of the i-th atom away from its equilibrium position R
of U (R) around its minimum value
X ∂U ∂ 2 U 1 X
U = U (R0 ) +
ui,z +
ui,z uj,r + . . . (2.23)
∂ui,z ui,z =0
2 i,j,z,r ∂ui,z ∂uj,r ui,z =0
i,z
where i, j label the N atoms of the molecule, whereas z, r for their Cartesian coordinates and U (R0 ) = Eαe is the electronic contribution at the minimum. The
first-order derivatives are zero since R0 is a minimum. We indicate the force constant matrix Φijzu (or Hessian matrix) as follows:
Φijzr
∂ 2 U =
∂ui,z ∂uj,r ui,z =0
(2.24)
It represents the r component of the force generated on the atom j, when the
atom i is displaced by a small amount in the direction z. For isolated systems, the
translation and rotational invariance conditions reduce the number of independent
elements in this matrix.
For vibration of small amplitude, it is safe to truncate the expansion (2.23) of
the total potential to second order. This is called harmonic approximation. The
17
2.3. MOLECULAR VIBRATIONS
classical Lagrangian corresponding to the quadratic potential
L=
1X
1 X
Mi u̇2i,z −
Φijzr uiz ujr
2 i,z
2 i,j,z,r
yields the following Lagrange equations
X
Mi üi,z +
Φijzr uj,r = 0.
(2.25)
(2.26)
j,r
This is a system of 3N coupled linear differential equations. Since we want to
include on the same footing also the case of a system containing different masses,
its convenient to scale the displacement coordinates ~u introducing a mass-weighted
displacement coordinates, defined as
p
(2.27)
q~i = Mi ~ui
The linear system (2.26) becomes, in terms of ~qi
X
q̈i,z +
Dijzr qj,r = 0
(2.28)
j,r
where D is the dynamical matrix of the system
1
Dijzr = p
Φijzr
M i Mj
(2.29)
Substituting a general oscillatory solution qi,z = Ai,z cos(ωt + φ) in (2.26) we get
X
(δij δzr ω 2 − Dijzr )Aj,r = 0
(2.30)
j,r
The problem reduces to a canonical algebraic diagonalization for the dynamical
matrix D. The non vanishing eigensolutions are those which satisfy the secular
equation
det(ω 2 I − D) = 0
(2.31)
The dynamical matrix is a real and symmetric 3N × 3N matrix and can be diagonalized by an orthogonal transformation (i.e. a basis change). The new set of
coordinates Qv which diagonalize the dynamical matrix are called normal coordinates of vibration, and defined by a linear transformation of displacements. The 3N
eigenvalues of D, ωv2 , are the squared frequency of the normal modes of vibration
for the system considered. Due to the translational and rotational invariance, it is
possible to show that for a system with N non-collinear atoms, only N osc = 3N − 6
18
CHAPTER 2. MOLECULAR PHYSICS
eigenvalues are nonzero (N osc = 3N − 5 for linear molecules). The six (five) zerofrequency modes correspond to the rigid translations and rotations of the system
[6]
In terms of the Qk the quadratic part of the potential energy expansion (2.23)
has the form
osc
N
X
U=
ωk2 Q2k
(2.32)
k=1
It can be easily shown that the kinetic energy retains its original quadratic form
when expressed in normal coordinates. The Hamiltonian, describing the nuclear
motion can then be written, in terms of the normal modes, as
H
vib
=
osc
N
X
k=1
1 2 1 2 2
P + ω Qk
2 k 2
(2.33)
where Pk = Q̇k are the conjugated moments to the normal coordinates Qk . In
Eq. (2.33) we have left out the translational and rotational contributions, which
decouple if we choose a reference system rigidly tied to the molecule [6].
The system is hence equivalent to a collection of N osc independent harmonic
oscillator. As anticipated, the quantum solution is immediately obtained from the
classical one by quantizing each harmonic oscillator in the standard way, promoting
Qk and Pk to operators with the canonical commutation rules, [Q̂k , P̂k0 ] = ih̄δkk0 . It
is convenient to rescale the normal coordinates
q by a non-canonical transformation
p ωk
1
Q̂k → Q̂k and
P̂ → P̂k with [Q̂k , P̂k0 ] = iδkk0 .
to dimensionless ones:
h̄
h̄ωk k
Then Hamiltonian (2.33) reads
N osc
H vib
1X
h̄ωk P̂k2 + Q̂2k
=
2 k=1
(2.34)
The usefulness of this form is that it yields naturally the phonon description of
the molecular vibration in terms of creation and annihilator operators, respectively
defined, for each normal mode, by
1
ak ≡ √ Q̂k + iP̂k ,
2
1
a†k ≡ √ Q̂k − iP̂k .
2
(2.35)
In this language, Eq. (2.34) can be rewritten
H
vib
=
osc
N
X
k=1
h̄ωk a†k ak
1
+
2
(2.36)
19
2.3. MOLECULAR VIBRATIONS
with nonnegative integer eigenvalues of the number operator a†k ak , that following
the standard notation of molecular spectroscopy, we shall indicate as vk . Hence, the
quantum many-body problem for the nuclei, if adiabaticity and harmonicity conditions are satisfied, can be solved exactly. The solutions of the Schrödinger equation
(2.15) for the vibrational motion are expressed in terms of the eigenvalues Evk and
the corresponding eigenstates ϕvk of the single normal modes Qk of frequency ωk ,
given by:
1
Evk = h̄ωk (vk + )
2
ϕvk (Qk ) = Nvk e
−
Q2
k
2
Hvk (Qk ),
(2.37)
N vk =
1
π 1/2 2vk v
k!
1/2
(2.38)
where Hvk (Qk ) are Hermite polynomials [11]. ϕvk (Qk ) are eigenstates of the number operators a†k ak with eigenvalues vk and can be interpreted as states containing
exactly vk “phonons” of frequency ωk . The complete eigenvalues are
Evvib
= Ev1 + Ev2 + . . . + EvN osc =
osc
N
X
1
h̄ωk (vk + ),
2
k=1
(2.39)
and the corresponding eigenstates
φv (Q) = ϕv1 (Q1 )ϕv2 (Q2 ) . . . ϕvN osc (QN osc ).
(2.40)
v indicates collectively the set of phonons’ quantum numbers v1 v2 . . . vN osc . The
state φv (Q) is then a state with v1 phonons of frequency ω1 , v2 phonons of frequency ω2 and so on. It is useful to observe that P
theoscground-state, for which all
1
quantum numbers are zero, has energy equal to 2 N
k=1 h̄ωk above the bottom of
the adiabatic potential well U (R0 ), which may be of considerable magnitude in
polyatomic molecules. This is just a quantum effect. Sometimes in this work we
use the more compact Dirac form for the total vibrational eigenstate φv (Q)
|vi = |v1 v2 . . . vN osc i
(2.41)
In the general cases, when we have large distortions and the harmonic approximation fails, higher-order terms of the potential expansion (2.23) are to be considered. They may be interpreted as interactions between the phonons. Collective
terms ∆Ev have to be added to the simple expression (2.39) for the total energy.
Moreover, the eigenstates will be linear combinations of the harmonic ones (2.40).
The general Hamiltonian, in terms of the normal modes Q, takes the form
N osc
1X
h̄ωk P̂k2 + Q̂2k + Gpp (Q̂)
H=
2 k=1
(2.42)
20
CHAPTER 2. MOLECULAR PHYSICS
where Gpp is a function of all the normal coordinates Q̂ globally accounting for the
anharmonic terms of the potential energy (2.23). Inverting relations (2.35)
1
Q̂k = √ (a†k + ak ),
2
i
P̂k = √ (a†k − ak )
2
(2.43)
we can rewrite Eq. (2.42) in terms of creators and annihilators
N osc
1
1X
†
h̄ωk ak ak +
+ Gpp ({a† }, {a})
H=
2 k=1
2
(2.44)
where with {a† } and {a} we compactly indicate all creators and annihilators. This
picture underlines the phononic character of the vibronic motion and the subscript
pp indicates the phonon-phonon character of the anharmonic interactions.
2.4
The full molecular Hamiltonian
Combining the results of the previous sections, we can rewrite the vibronic Hamiltonian (2.1) in a form which explicitly shows the approximations used to approach the
problem. Recalling Eqs. (2.21), (2.22) and (2.42), the adiabatic part of Hamiltonian
(2.1) reads
H
adi
=
X
a
N osc
a c†a ca
1X
h̄ωk P̂k2 + Q̂2k + Fee ({c† }, {c}) + Gpp (Q̂)
+
2 k=1
(2.45)
Here, the first term represents the single-electron contributions, the second term is
the harmonic phonon Hamiltonian, as a function of the normal modes of frequency
ωk , where it is implicitly understood that ωk , the equilibrium positions R0 and
the normal-mode coordinates Qk depend on the electron occupancies. The next
two terms, Fee and Gpp (Eqs. (2.22) and (2.42)), can be seen as corrections to
independent-electrons and independent-vibrations approximations: they describe
respectively electron-electron correlations terms and anharmonic phonon-phonon
interactions. The electron-phonon picture of the system can equivalently well be
expressed in terms of phonon creators and annihilators as in Eq. (2.44) or in terms
of the normal distortions Q̂, related to the creation operators by Eq. (2.43). In
this electron-phonon picture, the non-adiabatic terms, ignored until now, appear as
electron-phonon interactions which can globally be described by a function Iev of
the electronic creators and annihilators {c† }, {c} and of the normal distortions Q̂.
The Hamiltonian (2.1) can then be rewritten as
H = H adi + Iev ({c† }, {c}, Q̂)
(2.46)
2.5. SYMMETRY AND VIBRONIC STATES CLASSIFICATION
21
For example, a linear e-v interaction, like c†a cb Qv , describes a process in which a
phonon absorption or creation induces an electron to jumps from state b to state a.
In general the form of these interactions terms (this is true also for Fee and Gpp ) is
restricted by some symmetry property they have to satisfy, e.g. translational and
rotational invariance, point group symmetry. Electron-phonon terms are especially
relevant when the electronic and vibrational excitation energies are of comparable
size: typically, significant effects are observed only involving electrons states close to
the Fermi energy. In particular e-v couplings are important for orbitally degenerate
or almost degenerate electronic ground states, where inducing transitions between
different electronic levels is especially easy.
2.5
Symmetry and vibronic states classification
We conclude this chapter with some note about the importance of symmetry considerations to classify molecular states. Each molecule in its equilibrium configuration
may be characterized by a certain set of symmetry operations under which it is
invariant. This set forms the point symmetry group of the molecule and may be
composed mainly by three kinds of operations: rotations by an integer fraction of
2π about an axis, reflections with respect to a plane and reflections around a point
(space inversion) [12, 6].
In the Schrödinger equation for the electron motion (2.5), Ve = He − Te is
the potential energy that the fixed nuclei in their equilibrium position induce on
the electrons (2.6). Therefore Ve has the symmetry of the molecule in that particular equilibrium configuration. Thus, if a symmetry operation is carried out,
the Schrödinger equation for the electronic motion remains unchanged. As a consequence, the electronic eigenfunction for nondegenerate states can only be symmetric
or antisymmetric for each of the symmetry operations permitted by the symmetry
of the molecule in the equilibrium position. For degenerate states, the eigenfunction can only change into a linear combination of the other degenerate eigenfunctions such that the square of the eigenfunction in an equally populated statistical
ensemble of the degenerate states, which represents the electron density, remains
unchanged.
Different eigenfunctions may behave differently with respect to the various symmetry operations of a given point group; but in general not all symmetry elements
of a point group are independent of one another: only certain combinations of behavior of the eigenfunctions with regard to the symmetry operations are possible.
Such combinations of symmetry properties are the irreducible representations of the
point group, also called symmetry species or types. Each electronic eigenfunction
and therefore each electronic state, belongs to one of the possible symmetry types
of the point group of the molecule in its equilibrium position.
22
CHAPTER 2. MOLECULAR PHYSICS
If the ions were fixed in positions other than their equilibrium positions and if
the symmetry of the displaced positions is different from that of the equilibrium
positions, the species of the electronic wavefunctions would be different. However,
since there must clearly be a one-to-one correspondence between the electronic
eigenfunctions in the two conformations, we can still at least for small displacements
(vibrations), classify the electronic eigenfunctions according with their species at
the equilibrium conformation.
We must note, however, that in degenerate electronic states, for certain displacements from the equilibrium configuration, there may arise a splitting of the
potential surface, since in the displaced conformation the symmetry may be lower
and degenerate species may not exist [7].
As seen in Sect. 2.2 the manifold of electronic states is a tensor product of
the single-particle ones (2.20). In a first approximation, also the single-particle
wavefunctions are irreducible representations of the point group and we can classify the total eigenstates in terms of the symmetry of the occupied single-particle
levels. It is the same that happens in the standard treatment of many-electrons
atoms. The average potential acting on each electron is approximatively symmetric under rotations. In the Hartree-Fock approximation, single-electron levels are
typically labeled by quantum number j and mj corresponding to the total angular
momentum of single-particle levels and to its component along the z-axis: these
quantum numbers characterize an irreducible representations of the group of all
the 3-dimensional rotations. The symmetry of the total eigenstates is the result of
the composition of these single particle levels. In particular, concretely, all totally
symmetric closed shells give vanishing contributions and the symmetry properties
of the total eigenstates just depend on the partly filled outer shell.
Similar considerations hold for the phonons wavefunctions. The energy potential Uα (R) governing the nuclear motion Eq. (2.15), is invariant under the pointgroup transformations when the molecule occupies the equilibrium configuration R0 .
Again, distorted configurations generally break the equilibrium symmetry. However,
for small displacements, it can be easily shown that a symmetry transformation of
the coordinates induces a linear transformation of the displacements (a one-by-one
correspondence) [6]. Then, the single-mode wavefunctions given by Eq. (2.38), are
labeled by irreducible representations of the point group of the molecule. The symmetry of the total phonon eigenfunction is given, as for the electronic states, by
the composition of single-mode symmetry. Note that the N osc normal frequency
ωk satisfying the secular equation (2.31), are not necessarily all distinct. These
degeneracies reflect the degenerate representations of the point group and the components of one degenerate normal mode mix among each other upon a symmetry
transformation.
Hence, the adiabatic harmonic vibronic states (2.17) are naturally classified
through the irreducible representation of the point group they belong to, deter-
2.6. DYNAMIC JAHN-TELLER EFFECT IN MOLECULES
23
mined by the composition of the phononic and electronic characters.
In the general case, anharmonic interactions between phonons, electron-electron
interactions or non-adiabatic electron-phonon couplings do not lower the symmetry
of the molecule: the general solution (2.18), a linear combination of the symmetric adiabatic harmonic eigenstates shell, involves only vibronic states belonging to
the same representation of the point group of the equilibrium configurations. As
a familiar example, the spin-orbit interaction in a one-electron atoms breaks the
rotational symmetry in separate spin and orbital space, but it is invariant under
global rotations of both spin and orbital degrees of freedom: as a consequence, the
eigenstates of the interacting system are labeled by the total angular momentum
and projection quantum number j and jz .
For the subject of the present work, systems with degenerate or almost degenerate electronic ground state, where e-v couplings induce a spontaneous symmetry
breaking, are the most interesting. These system are the so-called Jahn-Teller
molecules to which we devote the final of this Chapter.
2.6
Dynamic Jahn-Teller effect in molecules
The Jahn-Teller effect (JT), first suspected by Landau in 1934 [13], takes place
in any case of (partly filled) degenerate electronic level in non linear molecules
and in complexes. It is due in last analysis to the very general tendency of any
system toward a closed shell. There is always some energy to gain by splitting
an electronic degeneracy (typically breaking the associated symmetry), and filling
the lowest split-off levels. Some energy has to be paid, of course: the splitting
is obtained by changing the geometric configuration of the ions, which move away
from their original equilibrium positions. The restoring energy, however, is typically
quadratic in the displacement from the potential minimum, while the leading term
in the splitting of a degenerate level, and thus the gain in electronic energy, is usually
linear in the perturbing distortion (at least until the split levels remain far apart
from the neighboring electronic levels): the balancing of these two terms brings the
molecule to a new minimum of the potential energy, in a distorted configuration.
It is customary to consider an expansion, with linear, quadratic, cubic. . . terms
in the distortion coordinates Q of the e-v interaction of Eq. (2.46). We refer to
neglect of all terms beyond the lowest one, as linear coupling approximation. In the
following we will indicate with g the Hamiltonian parameter tuning the strength
of the linear e-v coupling term. Usual second-order perturbation theory, applied to
degenerate state, yields a net ground state energy lowering proportional to g 2 ∆E
with a distortion proportional to g. Here ∆E is the energy difference between the
ground state and the closest interacting excitation, typically a vibrational quantum
h̄ω.
24
CHAPTER 2. MOLECULAR PHYSICS
1
0.1
0
-0.1
-0.2
-0.3
0.5
0
-1
-0.5
-0.5
0
0.5
-1
1
Figure 2.1: A pictorial representation of the lowest adiabatic potential surface for linear
JT coupling, showing six equivalent minima.
If, as in most cases, the degeneracy was originally due to an exact symmetry
of the system, then the symmetry-breaking distortion has to choose to take place
in one of the possible symmetry-equivalent directions. If the vibrations can be
regarded as classical, then any energy barrier among the system-equivalent minima
freezes the molecule in one specific symmetry-broken configuration, around which it
performs low-energy oscillations. This picture is known as static JT effect: thermal
effects may allow jumps to the other equivalent minima of the potential surface.
Static JT systems occur whenever JT distortions are very large.
Far more attractive phenomena arise when a quantum mechanical description
of the motion of the ions is necessary. The minima of the adiabatic potential can
now communicate with one another through quantum tunneling. Thus, instead of
having a symmetry broken distorted ionic configuration, the ground state resonates
among the equivalent minimum-energy distortions, thus restoring the original symmetry. Tunneling is present in any system, but when its reciprocal frequency is
large compared to the time scale of any experiment, a static picture is satisfactory.
On the contrary, when the energy barriers separating different minima are low or
even absent, a full quantum description of the state of the system is essential to
its correct understanding. The denomination dynamical JT (DJT) applies in such
cases.
The name DJT only applies when the vibration modes interacting with the
degenerate electronic state are themselves degenerate. The only trivial effect of
JT coupling to non-degenerate vibrational coordinates is to shift the equilibrium
2.6. DYNAMIC JAHN-TELLER EFFECT IN MOLECULES
25
Figure 2.2: A schematic representation of the e ⊗ E adiabatic potential surface for linear
JT coupling, showing a flat JT valley, or trough.
position of the oscillator, without splitting the electronic degeneracy. Degenerate
vibrations instead often split the degeneracy of the electronic orbitals. This fact
introduce a generalized concept of adiabatic (BO) potential surface. For zero distortion, the electrons can equivalently well occupy any one of the degenerate orbitals.
When the levels are split, we get a multi-valued adiabatic potential surface, with
a conic intersection of several sheets at zero distortion (see Fig. 2.2). For weak JT
coupling (g → 0), the terms mixing the different almost degenerate BO sheets are
crucial, all the eigenstates involve coherent mixtures of different electronic states
and distortions: a traditional BO description, in term of the lowest sheet only, is
completely inaccurate. Instead, for strong JT distortion (g → ∞), the so-called
adiabatic limit, the separation of different electronic levels is large, thus most spectral features can be extracted from the study of the lowest BO sheet (see Sect. 2.1),
treating electronic excitations from the lowest to higher adiabatic sheets as unlikely
virtual process.
The structure of the set of absolute minima of the BO potential of a JT-coupled
system is crucial in the determination of its low-energy spectral properties for intermediate to strong JT coupling. The minima are usually a discrete set of isolated
point, as depicted in Fig. 2.1 (for a beautiful example see Ref. [14]), related by the
26
CHAPTER 2. MOLECULAR PHYSICS
symmetry operations of the molecular point group, and joined by “energetically
cheapest” paths (i.e. path such that at every point on them the potential energy increases in all the direction perpendicular to the path) passing through saddle points
of the potential surface. In many cases, such as the linear e ⊗ E and t ⊗ H models,
the set of potential minima becomes a continuous manifold, called JT valley or
trough as in Fig. 2.2. It is exactly flat only in the ideal case of perfectly linear JT
coupling. Higher-order coupling in practice reintroduce always the isolated minima
- saddle point structure, but they are often small corrections, and a description in
terms of a flat valley yields quantitative results.
The vibron spectra in the case of many perfectly isolated minima (adiabatic
limit) is characterized by vibrational spectra having the additional degeneracy given
by the many possible equilibrium configurations. As the barriers become finite,
these vibronic state give rise to tunnel-split states carrying a label of the molecular
point group, increasingly separated for increasing tunneling amplitude. When the
saddle-points vanish and we get a flat JT valley, the ions undergo a free-rotation-like
motion along this locally sphere-like manifold plus harmonic vibration orthogonal to
it. Vibronic states correlate continuously as the interminima barriers are reduced.
The next chapter is devoted to apply these concepts to C60 ions which, for their
high symmetry, are well known DJT systems. In particular, the positive ions, which
are the main subject of this work, realize the so called h ⊗ (G + H) DJT system,
which is characterized by isolated minima. The following Chapter discusses the
relevant model for degenerate electron-phonon interaction (DJT) in C+
60 .
Chapter 3
The model for dynamic
Jahn-Teller in C60
The C60 molecule has a completely filled highly degenerate outer orbital and photoemission generates C+
60 cations with electronically degenerate ground state, for
which e-v coupling induces DJT distortion. In this chapter we review only a few
basic aspect of C60 , namely those useful for studying e-v couplings in C60 ions.
In particular, with an emphasis on the icosahedral molecular symmetry, we give a
brief introduction to the electronic (Sect. 3.2) and vibrational (Sect. 3.3) structure
of C60 , and describe the JT coupling in C+
60 with the model of Sect. 3.5. The aim
is to provide the framework for the DJT model we use for our calculation, setting
up the notations used throughout this work. For more complete reviews of C60
properties we suggest Refs. [15, 16, 17, 18].
3.1
The fullerene molecule
The 60 carbon atoms of fullerene are all threefold coordinated and arranged as a
truncated icosahedron (Fig. 3.1), a regular, roughly spherical cage, of about 7 Å
in diameter, [19]. The cluster may be thought of as a small piece of graphite
sheet wrapped to spherical shape. The regular hexagonal structure of graphite is
distorted, and 12 five-membered rings intercalate 20 six-membered ones, in such an
ordinate way that all atoms are equivalent, each shared among one five-membered
ring and two six membered rings, as shown in Fig. 3.2. The 90 chemical bonds
divide into two classes: 30 of them belong to six membered rings only, the others
are in common between the two kinds of rings. They have a different length, with a
mean value of about 1.44 Å. The σ-bonding sp2 graphite orbitals still constitute the
backbone of molecular binding. Spherical curvature alters their character to sp2.28
[20]. The σ-bonding orbitals range from several eV’s to a few tens of eV below the
27
28
CHAPTER 3. THE MODEL FOR DYNAMIC JAHN-TELLER IN C60
Figure 3.1: A truncated icosahedron
vacuum, the antibonding states lying +10 eV and higher above the vacuum [21].
The chemically-active electronic states, mainly of π-type, are those derived from
the pz carbon orbital (actually of hybrid s0.09 p nature [20]) aligned in the radial
direction. These orbitals concur to form a set of delocalized molecular orbitals,
providing an energy gain which accounts for extra molecular stability. In fact,
despite the formally large number of unsaturated bonds, C60 is a pretty stable
allotropic form of carbon, both as an isolated molecule and as a fcc solid. The next
section contains a few more details about the π-bonding/antibonding molecular
orbitals, characterizing the electronic structure.
The overall molecular symmetry group is the icosahedral group Ih , the largest
point group in 3D (except for axial groups), composed by 120 elements. Ih is the
product group of I × i, where i is composed by the identity and the space inversion, and I contains 60 rotations organized in the following classes [6]: the identity,
12 C5 , 12 C52 (through the center of opposite pentagonal rings), 20 C3 (through the
center of opposite hexagonal rings), and 15 C2 (through the middle of the bonds
shared between two hexagonal rings). The great richness of this group is directly
related to the large degree of symmetry carried by this unique molecule, appearing
in particular in the complete equivalence of all its 60 atoms. The large symmetry is
also reflected in the large degeneracy of the group’s irreducible representation [6]:
A (1-dimensional), T1 , T2 (3-dimensional), G (4-dimensional), H (5-dimensional).
In Ih , the character of inversion adds a g/u subscript for even/odd representation,
but leaves the degeneracies unchanged. As discussed in Sect. 2.5, all of (electronic,
vibrational, vibronic) molecular states and excitations are basis of irreducible representations: degeneracies up to fivefold are therefore very common. This makes C60
a promising system for novel Dynamic Jahn-Teller structures [16, 22, 23, 24, 25, 26].
3.2. THE ELECTRONIC STRUCTURE
29
Figure 3.2: A 3-dimensional view of the C60 molecule
This large symmetry, however, is also delicate. For example, isotope substitution
of one carbon atom is enough to reduce Ih to simple bilateral reflection Cv . Due to
1.10% isotope abundance of 13 C in natural carbon, only 51.5% of fullerene is pure
12
C60 . However, isotope substitution induces very small splittings of vibron modes
and electronic orbitals, and can be safely neglected for many purposes. 13 C12 C59 ions
have almost-degenerate ground states still affected by Jahn-Teller effect (Sect. 2.6)
in essentially the same way as 12 C60 .
C60 forms a Van der Vaals fcc insulating solid (lattice parameter 14.17 Å) of
loosely bound individual molecules. It sublimates at approximately 800 K, without passing through a liquid phase. Doping C60 can form ionic materials (fullerides), showing novel and exciting properties such as superconductivity, organic
magnetism, correlated Mott-insulating states.
3.2
The electronic structure
As anticipated, in this section we concentrate on the π orbitals and electrons, which
are those in the chemically relevant region of the spectrum. We use a one-electron
orbital picture, neglecting e-e effects at this stage. Although in principle unjustified
(the bare Coulomb repulsion of two electrons separated by a fullerene radius is
about 3 eV), the one electron picture is, as always, pretty useful (Sect. 2.2).
Many approaches have been taken to the electronic structure of C60 , such as
tight-binding [27, 20, 21], local density approximation [21], or renormalization group
methods, yielding increasing degrees of quantitative accuracy.
30
CHAPTER 3. THE MODEL FOR DYNAMIC JAHN-TELLER IN C60
Figure 3.3: A particle-on-a-sphere schematic representation of the electronic levels of C 60 .
The HOMO and LUMO, originating from the L = 5 orbitals, are evidenced. (adapted
from Ref. [28])
For the purpose of this chapter, we review a more intuitive method, of a particleon-a-sphere, giving a simple qualitative description of the electronic structure. It
treats the π electrons as independent particles sliding on a spherical shell of radius
R, mimicking the attractive potential generated by the carbon ions as if they were
smeared on this shell [28]. As long as icosahedral perturbations can be neglected,
the single-electron π-states are atomic-like spherical harmonics YLm , with energy
E(L) =
L(L + 1) h̄2
,
2me R2
(3.1)
proportional to the angular momentum of rotation around the center of the sphere.
Hence, 2 + 6 + 10 + 14 + 18 = 50 out of 60 π electrons (1 per atom) of neutral C 60
fill completely the molecular orbitals up to L = 4, whereas the remaining 10 are
3.3. THE VIBRATIONAL MODES
31
left in the L = 5 shell, which is therefore only partly filled.
An icosahedral perturbation [28] is subsequently added accounting for the actual
positions of the ions. The lowest L = 0, 1, 2 orbitals fit exact icosahedral labels ag ,
t1u , hg . The icosahedral field generated by the cage splits the L > 2 spherical states
into icosahedral representation, as shown in Fig. 3.3. In particular the splitting
of the partly filled outer shell, L = 5 → hu + t1u + t2u , generates a closed shell
configuration. In accord with previously mentioned theoretical approaches and with
experiment, the completely-filled Highest Occupied Molecular Orbital (HOMO) has
hu symmetry, and the Lowest Unoccupied Molecular Orbital (LUMO) is t1u . The
HOMO-LUMO gap opened in the L = 5 shell is about 2 eV. A direct experimental
measure of this gap is not easy, therefore that is just an indicative value. More
accurately known are the values of the ionization potential, 7.58 eV, and the electron
affinity, 2.65 eV.
This description is certainly an approximation, but it provides a simple mnemonic,
fairly accurate picture of molecular orbitals of fullerene. We are mainly interested
in the HOMO, to which photoemission subtracts electrons (or equivalently adds
holes), producing positive ions. Given the large size of C60 , it is reasonable to assume that the inner orbitals feel negligible change upon ionization of the molecule,
so that C+
60 has generally the same shape and bonding as neutral C60 , the effects of
charging being mainly restricted to the one-particle effects induced by the hole in
the HOMO.
3.3
The vibrational modes
In a complex molecule as C60 , vibrations play an important role, and a complete
understanding of the spectral features cannot leave them out (Sect. 2.3).
C60 has 174 vibrational degrees of freedom, but thanks to symmetry-induced
degeneracy and selection rules, the vibrational spectra show relatively few peaks,
and nowadays there is general agreement on their identification. The cleanness of
the observed vibrational spectra pays the price of leaving most vibrational modes
inaccessible to direct infrared and Raman spectroscopy. In particular only the four
T1u dipolar modes are seen in infrared, while the two Ag and eight Hg modes have
the correct symmetry for Raman scattering. Neutron scattering experiments [29]
are sensitive to all the modes, but with rather poor resolution. Computations are
therefore useful , for getting a global view of the vibrational spectrum.
Many routes to the calculation of vibron eigenfrequencies and normal modes
have been pursued, either based on force fields fitted on the experimentally accessible data [29, 31, 32], or based on ab initio computations of the molecular structure
[3, 33, 34, 35]. The agreement among different computations is surprisingly poor.
Phenomenological models have typical deviations of the order of the 10 meV, while
32
CHAPTER 3. THE MODEL FOR DYNAMIC JAHN-TELLER IN C60
Figure 3.4: Spherical resolution of the vibrational spectrum of C 60 . The modes are
organized according to the three series ω 1/2/3 . The spherical parent L is indicated before
the icosahedral label [From Ref. [30]]. The highlighted modes are the active modes which
couple to the HOMO.
3.4. E-V COUPLING: SYMMETRY CONSIDERATIONS
33
typical discrepancies of good ab-initio calculations are on the order of 1 meV, well
above nowdays’ experimental resolution. All the calculations in this work are based
on the ab-initio frequencies and normal modes of Ref. [3]. We report them for convenience in Table 3.1, together with the experimental frequencies. We do not use the
measured values because accurate mesurements for the Gg modes, are unavailable.
Like for the electronic structure, here we content ourselves with a qualitative
understanding of the vibrational structure. The schematic logical framework that
we briefly summarize here exploits the analogy of the C60 cage with a hollow elastic
sphere [30]. There are three classes of eigenmode solutions of a homogeneous shell
with a stretching and a bending constant. The first class contains levels of parity
(−1)L , with L = 0, 1, 2, 3, . . .; the levels in the second class have the same parity, but
start off at L = 1; in the third series the parity is reversed (−1)L+1 , and they also
start at L = 1. The vibrations of the third class have purely tangential character,
while those of series 1 and 2 have mixed radial-tangential nature (series 1, at higher
energy involve mostly stretching, series 2 mostly bending). The zero-energy L = 1
states of series 2 and 3 are the modes of rigid translation and rotation of the sphere.
Of course, when the homogeneous balloon is replaced by discrete 60 atoms bound
together by the molecular adiabatic potential, the infinite set of spherically symmetric eigenmodes is cut off to a finite number of modes, now labeled by Ih representation. L = 0, 1, 2 states have the icosahedral counterparts in Ag/u , T1u/g , Hg/u
respectively: the first label corresponds to series 1 and 2. The second labels the
third series, which has inverted parity. As for the electronic orbitals, the states with
L > 2 are split, according to rules given in TABLE III of Ref. [30]; for example
both L = 3 odd-parity spherical modes generate a T2u and a Gu icosahedral mode.
The explicit eigenmodes, computed for example by force field methods, can be
easily analyzed in term of the spherical basis, to obtain their parentage in terms of
hollow-sphere modes [30]. The three series are readily identified in the three columns
of Fig. 3.4, labeled ω1 , ω2 , ω3 . Inter-mode mixing is present, but most vibrons have
a well defined parentage, with few exceptions. At any rate, this spherical picture
remains generally a valid and useful conceptual scheme.
3.4
Electron-vibration coupling: symmetry considerations.
Photoemission from C60 produces a molecular ion, C+
60 , with a hole in the fivefold degenerate hu HOMO. The adiabatic approximation (Sect. 2.1) fails, and nonadiabatic electron-phonon interactions have to be included (Sect. 2.4). JT effect
takes place and, according to the discussion of Sect.2.6, the molecule immediately
starts to distort into one of the six symmetry-equivalent static minima, calculated
34
CHAPTER 3. THE MODEL FOR DYNAMIC JAHN-TELLER IN C60
mode
symmetry
Λj
Ag 1
Ag 2
Gg 1
Gg 2
Gg 3
Gg 4
Gg 5
Gg 6
Hg 1
Hg 2
Hg 3
Hg 4
Hg 5
Hg 6
Hg 7
Hg 8
Raman
h̄ωΛj
h̄ωΛj
DFT-LDA calculation
h̄ωΛj
gΛj
[cm−1 ]
[cm−1 ]
[meV]
496
1470
271
437
710
774
1099
1250
1428
1575
500
1511
483
567
772
1111
1322
1519
261
429
718
785
1119
1275
1456
1588
62.0
187.4
59.9
70.3
95.7
137.8
163.9
188.4
32.4
53.2
89.0
97.3
138.7
158.0
180.5
196.9
[dim.less]
αΛj
[rad]
0.0591
0.2741
0.7567
0.1024
0.8003
0.6239
0.2277
0.4674
3.0417 -0.0015
1.2234 0.5250
0.9950 1.5605
0.7842 -0.0394
0.2212 1.3361
0.5194 0.4889
0.9616 0.4905
0.8686 -0.5430
Table 3.1: Computed vibrational eigenfrequencies of C 60 , and e-v linear coupling parameters for the hu HOMO [3]: comparisons with experimental values of the Raman-active
vibrational modes Ag and Hg [29, 31, 36].
to be of symmetry D5 [3, 37].
For a Hamiltonian description of e-v coupling, it is customary to consider an
expansion of the electron-phonon interaction of Eq. (2.46), with linear, quadratic,
cubic. . . terms in the distortion coordinate. For our calculation we use a linear
harmonic JT model including only the leading term. We also neglect anharmonic
terms in the distortions (see Eq. (2.44)). Several theoretical papers [37, 25, 26, 38]
formulate the so-called hu ⊗ (Hg + Gg + Ag ) JT problem, describing a hole in
molecular C60 . In this section we highlight the features of this model, and set up
explicitly the notations we use in the present work.
Basic symmetry considerations are crucial to determine the form of the coupling
Hamiltonian, for fullerene ions, and in general.
The e-v coupling operator must, on general grounds, be a scalar, i.e. a totally
symmetric representation of the group. In the C60 case it must be Ag (or L = 0, if
we enlarge to the full spherical group). The general structure of the linear coupling
3.4. E-V COUPLING: SYMMETRY CONSIDERATIONS
operator in tensor notation is
i(Ag )
X h
e−v
(γel )
(γel ) (γ)
(γvib )
H
∝
D
×D
,
×D
s
35
(3.2)
γ
(γ)
where · ⊗ ·
represent the symmetric coupling of two tensors to a representation
(γ)
D in standard group-theoretical notation.
In particular, the symmetric product of a hu icosahedral representation with
itself results
{hu ⊗ hu }s = ag ⊕ hg ⊕ (gg ⊕ hg ),
(3.3)
(2)
(γ=0,2,4)
like in spherical symmetry D × D (2)
, with the correspondence D (0) → ag ,
(2)
(4)
D → hg and D → (gg ⊕ hg ). The modes interacting with the HOMO are
therefore the Raman-actives ones, 2 Ag and 8 Hg , plus the 6 Gg modes, highlighted
in Fig. 3.4: the fact that the Hamiltonian is entirely determined by symmetry (up
to a constant) explains the name hu ⊗ (Hg + Gg + Ag ) model. Although this O(3)
picture is suggestive [30], clearly a quantitative description of C+
60 ions had better
involve icosahedral symmetry from the beginning.
In the icosahedral group, indeed, in the hu ⊗ hu tensor product of Eq. (3.3),
the hg representation appear twice. This reflects the non simple reducibility of the
icosahedral symmetry group. Accordingly, Butler [39] provides two orthogonal sets
of Clebsch-Gordan (CG) coefficients
H
m [r]
Cµ,ν
≡ hh, µ; h, ν|H, mi[r],
r = 1, 2
(3.4)
which couple an h electronic state (quadratically) with an H vibrational mode
(linearly) to give a scalar. Each set of coefficients is identified by a multiplicity
index r = 1, 2. Since the two H states labeled r = 1, 2 are indistinguishable
symmetry-wise, the choice of these two sets of coefficients is perfectly arbitrary, as
long as they are kept orthogonal to each other. This arbitrariness is the source
of the different notations taken in the literature of this field. Here, we stick to
Butler [39], which is basically equivalent to Ceulemans’ convention [40]. Also, we
label the states within a degenerate multiplet by the labels of the subgroup chain
Ih ⊃ D5 ⊃ C5 . For brevity, we indicate only the C5 index m (m = 0 for a,
m = ±1, ±2 for g and m = −2, . . . , 2 for h states) in the labeling of states since, for
the representations relevant to our problem, the D5 label is just the absolute value
of m. In particular, we use µ, ν to label the orbital multiplet hu , and m to label
the multiplets corrisponding to normal modes.
Given the tabulated CG coefficients, to describe the coupling of an actual vibrational mode of an actual icosahedral molecule, it is necessary for generality to
consider the linear combination
H
m
Cµ,ν
(α) ≡ cos α
H
m [1]
Cµ,ν
+ sin α
H
m [2]
Cµ,ν
(3.5)
36
CHAPTER 3. THE MODEL FOR DYNAMIC JAHN-TELLER IN C60
m
of the two sets. The coefficient h Cµ,ν
(α) coincides with Butler’s r = 1 and r = 2
π
values for α = 0 and α = 2 respectively. Different values of α can be compared with
the conventions of previous authors [25, 26, 37]. The α-dependence of these CG
coefficients indicates that belonging to the hg group representation in icosahedral
symmetry does not determine completely the form of the JT coupling. The mixing
angle α is needed for that. In the present case of fullerene, each Hg vibrational
mode is thus characterized not only by its frequency and scalar coupling (as the Ag
and Gg modes), but also by its specific mixing angle −π/2 ≤ α ≤ π/2, reported in
Table 3.1.
3.5
The hu ⊗ (Hg + Gg + Ag ) model
The linear JT Hamiltonian for the hu ⊗ (Hg + Gg + Ag ) model, besides the electronvibration interaction term He−v itself, contains also a single-electron Hamiltonian
H0 , and a vibrational Hamiltonian Hvib as follows:
H = H0 + Hvib + He−v .
(3.6)
The individual terms can be written as
X
H0 = c†µ cµ ,
(3.7)
Hvib =
(3.8)
µ
X h̄ωΛj
jΛm
He−v =
2
PΛjm
+ Q2Λjm ,
2
X k Λ gΛj h̄ωΛj X
jΛ
2
µνm
Λ
Cµm−ν (αΛj ) QΛjm c†µ cν .
(3.9)
Here represents the energy position of the HOMO level where electrons are photoemitted from. µ, ν, m label the components within the degenerate multiplets. j
counts the phonon modes of symmetry Λ (j = 1, 2 for Λ = Ag , j = 1 . . . 6 for Λ = Gg ,
j = 1 . . . 8 for Λ = Hg ). ωΛj are the frequencies of the active modes of C60 , listed
in Table 3.1. gΛj are the parameters tuning the strength of the linear e-v coupling,
m
also listed in Table 3.1. Λ Cµν
(αΛj ) are Clebsch-Gordan coefficients for coupling the
†
two hu fermion operators cµ to phonons of symmetry Λ [39]. Clearly αΛj is relevant
for Hg vibration only, according to Eq. (3.5). QΛjm are the dimensionless molecular normal-mode vibration coordinates (measured from the adiabatic
pequilibrium
configuration of neutral C60 , in units of the length scale x0 (ωΛj ) = h̄/(ωΛj mC )
associated to each harmonic oscillator, where mC is the mass of the C atom), and
PΛjm the corresponding conjugate momenta (Sect. 2.3). Finally, to keep the notation as simple as possible, we omit all spin indexes and sums, because linear JT
3.5. THE HU ⊗ (HG + GG + AG ) MODEL
37
coupling is a purely orbital phenomenon, completely diagonal in spin, and spinorbit, exceedingly small in C60 [41], is neglected throughout.
In our calculations we adopt the numerical values of the e-v coupling parameters
gΛj , and αΛj listed in Table 3.1, obtained from first-principles Density Functional
Theory (DFT) electronic structure calculation of Ref. [3]. A second DFT calculation
[42], based on a different functional, reports couplings that are similar on the whole
to those of Table 3.1.
Finally, this model can equally well describe the JT effect of the LUMO, relevant for negative C60 ions, and indicated by t ⊗ (A + H). The only modifications
involve eliminating the Gg modes, replacing the CG coefficients with suitable ones,
extending the µ/ν sums from -1 to 1 and interpreting the c† operators as creators
of electrons instead of holes. For the t ⊗ (A + H) JT, the product decomposition is
unique, thus no r index is necessary.
The e-v couplings gΛj in Eq. (3.9) are dimensionless, measured in the units of
the corresponding harmonic vibrational energy quantum h̄ωΛj . Modes (e.g. the
second Gg mode) with gΛj 1 are thus only weakly coupled; conversely, modes (in
particular the lowest Hg mode) with a large gΛj > 1 are strongly e-v coupled. The
1
1
1
numerical factors k Ag = 5 2 , k Gg = 54 2 , k Hg = 1 for the HOMO, and k Ag = 3 2 ,
1
k Hg = 6 2 for the LUMO are introduced for compatibility with the normalization
of the e-v parameters in Ref. [3]. The coupling of Table (3.1) are in the mediumcoupling regime, for which neither a perturbative approch nor the classic adiabatic
limit give good results: it is a regime characteristic of DJT-effects (Sect. 2.6).
The symmetric form of the Hamiltonian H (3.6) grants that its vibronic eigenstates belong to some Ih symmetry species. As discussed in Sect. 2.6, in the distorted configuration, symmetry is recovered by resonating tunneling among the
equivalent D5 minima.
In the single-partcle H0 (3.7), the hole counterpart of Eq. (2.21), the relaxation
of the inner shells induced by the presence of the hole is implicitly accounted for
by the coupling costant g [3]. Moreover, for a single-hole C+
60 , also intrashell holehole [43] interaction terms (2.22) are irrilevant. The only real approximation in this
terms is to neglect the dependence of the normal frequency ωΛj , of the symmetric
configuration, on the numbers of holes. In addition, we neglect anharmonic contributions (Eq. (2.42)) in the vibrational Hamiltonian Hvib (3.8) (partly justified by
the rigidity of C60 ), and the higher-order non-linear terms in the coupling Hamiltonian He−v (3.9). The approximations made should however capture the important
physics of the problem we address here, within the simplest possible model.
We complete the theoretical framework of this work in the next Chapter, where,
starting from the Fermi’s Golden Rule, we derive a useful mathematical expression
for the photoemission spectrum of C60 , suitable for the numerical approch presented
in final chapters.
38
CHAPTER 3. THE MODEL FOR DYNAMIC JAHN-TELLER IN C60
Chapter 4
The photoemission spectrum
4.1
Fermi’s Golden Rule
If a molecular system, in some initial quantum state Ψi , is acted upon by an electromagnetic perturbation of frequency h̄ω, represented by some energy perturbation
operator VI , the spectral response function, representing the rate of transition to the
final states is given by Fermi’s Golden Rule [44]
Ii (h̄ω) =
2π X
|hΨf |VI |Ψi i|2 δ h̄ω − (Ef − Ei )
h̄ f
(4.1)
where the summation is carried out on all possible final states Ψf . Ψi and Ψf , in
general, are many-body eigenfunctions of the system Hamiltonian, of energy Ei and
Ef respectively, describing both electronic and ionic motion. Eq. (4.1) is a semiclassical approach, treating the electromagnetic field classically. For simplicity,
we use discrete indexes only, but, in general, the sum of Eq. (4.1) involves also
unbound final states for which the sum is replaced by a suitable integration. The
spectrum is then composed by a sequence of “sticks” placed at energy Ef − Ei
with relative strength |hΨf |VI |Ψi i|2 , plus the continuum of unbound final states
(Chap 1). In practice, experimental resolution will always broaden the δ-function
sticks to continuous structures.
Obviously, different experiments highlight different spectral features. Many
techniques have been developed, to investigate different details of the spectrum.
The precise interpretation of Eq. (4.1) depends on the character of the experiment. For example, in optical absorption experiments, spectral features appear at
energy, coinciding with the energy separation Ef − Ei between the final and initial molecular states. Promotions occur when the photon energy h̄ω matches any
energy separation, provided that the matrix elements of the perturbation is nonvanishing. Instead, in photoemission, the final state is an unbound state of the
39
40
CHAPTER 4. THE PHOTOEMISSION SPECTRUM
ionized molecule and of the emitted electron and Ef = Efion + E e . To investigate
the molecular features of the spectra Efion , it is sufficient to measure the energy of
the emitted electron, as discussed in Sect 1.2 and 4.3 below. If before excitation we
have not a single Ψi but a statistical ensemble
states {Ψi } with probabilP of initial
ity ρi , in terms of the density matrix ρ = i ρi |Ψi ihΨi |, the total spectrum is the
weighted average
X
I(h̄ω) =
ρi Ii (h̄ω).
(4.2)
i
Eq. (4.2) has Eq. (4.1) as a special case at zero temperature for a non-degenerate
ground state Ψi=0 , which we can regard as the well-determined initial state of the
molecule. At finite temperature, the molecules occupy the thermally accessible
vibrational levels according to the Boltzmann distribution ρi = Z −1 e−βEi , and the
spectrum becomes:
X
X
e−βEi
(4.3)
I(h̄ω) = Z −1
e−βEi Ii (h̄ω), Z =
i
i
Thermal effects, in general affect the shape of spectrum, typically introducing some
amount of broadening. For a degenerate ground state, even at T = 0 one has to
average over the contributions of all the d degenerate components, as in Eq. (4.2),
with ρi = 1/d.
4.2
Green’s function approach
The resolvent of the Hamiltonian H is defined as
G(z) =
1
z−H
(4.4)
where the real energy z = E + i is accompanied by an infinitesimal positive imaginary part .
A diagonal matrix elements of G(z), Gii is known as Green’s function. A most
useful property of the Green’s function is its connection with the spectral function
Ii of Eq. (4.1): consider in fact the diagonal matrix element of the resolvent, on the
state VI Ψi of Eq. (4.1) produced by the electromagnetic perturbation on the initial
state:
1
VI |Ψi i,
(4.5)
Gii (h̄ω + i) = hΨi |VI†
h̄ω + i − H + Ei
where we have shifted z = h̄ω → h̄ω + Ei for direct comparison with Eq. (4.1) It is
possible P
to put in evidence some properties of the resolvent inserting the unit operator I = f |Ψf ihΨf | into Eq. (4.5), where Ψf form a complete set of eigenfunctions
41
4.2. GREEN’S FUNCTION APPROACH
of the Hamiltonian H with eigenvalues Ef . We have
Gii (h̄ω + i) = hΨi |VI†
=
X
f
=
X
f
X
m
|Ψf ihΨf |
|hΨi |VI |Ψf i|2
|hΨi |VI |Ψf i|2
1
VI |Ψi i =
h̄ω + i − H + Ei
1
=
h̄ω + i − (Ef − Ei )
h̄ω − (Ef + Ei ) − i
2
h̄ω − (Ef − Ei ) + 2
(4.6)
From Eq. (4.6), we see that for any > 0, Gii (h̄ω + i) is analytic and its imaginary
part is negative (Herglotz property). Close to the real energy axis, Gii (h̄ω) exhibits
poles at Ef − Ei − i, the difference between the eigenvalues of H and the energy
of the initial state Ψi . As a consequence, for small enough , the real part of
Gii (h̄ω) exhibits characteristic positive-negative oscillations around Ef − Ei , while
the imaginary part exhibits negative peaks of Lorenzian shape centered around
Ef − Ei ; in the limit → 0+ each Lorenzian narrows into a δ distribution:
lim+
→0
1
= δ h̄ω − (Ef − Ei ))
π h̄ω − (Ef − Ei ) 2 + 2
(4.7)
Comparing Eq. (4.1) and Eqs. (4.6),(4.7) , we relate the spectrum to the Green’s
function:
2
Ii (h̄ω) = − Im Gii (h̄ω + i)
(4.8)
h̄
The total spectrum at finite temperature can then be expressed, combining
Eq. (4.8) and Eq. (4.3), as
I(h̄ω) = −
2 X e−βEi
Im Gii (h̄ω + i0+ ),
h̄ i
Z
Z=
X
e−βEi
(4.9)
i
Relations (4.8) and (4.9) hold regardless the fact that the energy spectrum of
H is discrete or continuous. In practice the finite experimental energy resolution
may be simulated by a small but finite , which replaces the δ functions at the poles
with Lorenzians.
We conclude that calculating the spectrum means basically to find the poles
Ef − Ei and residues |hΨi |VI |Ψf i|2 of the Green’s function (4.6). The usefulness of
expressions (4.8) and (4.9) is that these can be treated numerically through standard
algorithms such as the Lanczos method and the continued fraction, summarized in
Chapter 6.
42
4.3
CHAPTER 4. THE PHOTOEMISSION SPECTRUM
Photoemission: the sudden approximation
In photoemission, a radiation of known frequency ω ionizes the molecule, generating
the final unbound state of a free electron plus a molecular ion (Chap. 1 ). The final eigenfunctions Ψf are product of the wavefunctions Ψfion , describing the ionized
molecule, times the free-particle wavefunctions of the emitted electron, ψ~ke , characterized by its wave vector ~k. In angular-integrated photoemission, the electron
detector selects a well defined energy E|e~k| =
h̄2 |~k|2
2me
of the outcoming photoelectron.
Efion
Indicating with
the final energy of the molecular ion, Eq. (4.1) specializes to
photoemission as follows:
Ii (E) =
2π
h̄
X
f,~k: E e~ =E e
|k|
|hΨfion ψke~0 |VI |Ψi i|2 δ h̄ω − E|e~k| −(Efion − Ei )
| {z }
E
(4.10)
Eq. (4.10) explicitly shows what is actually measured in a photoemission experiment, i.e. the binding energy E = h̄ω − E|e~k| by fixing the kinetic energy E|e~k| = E e
of the emitted electron.
Having clarified these points, it is better to rewrite Eq. (4.10), more compactly,
in the familiar form (4.1), as
Ii (E) =
2π X
|hΨf |ṼI |Ψi i|2 δ E − (Ef − Ei )
h̄
(4.11)
f
where from now on we refer to Ψi and Ψf respectively as the initial neutral molecular state and the final ionized molecular state, with energy Ei and Ef , including
implicitly the angular integration on the free electron final states in the perturbation
Hamiltonian ṼI
Moreover, we suppose that radiation pulls up the electron instantaneously, in a
time much shorter than the characteristic vibrational times, so that the ions have
no time to relax during the electronic transition. Immediately after photoemission,
C+
60 starts evolving from its initially symmetrical configuration, characteristic of
neutral C60 , and relaxes toward equilibrium (sudden approximation).
In the sudden approximation the perturbation Hamiltonian ṼI is the purely
electronic operator
X
C(h̄ω, α, |~k|) c†αµ ,
(4.12)
ṼI =
αµ
where α labels the molecular orbital, and µ its degenerate component. Each term in
Eq. (4.12) creates a hole in a specific orbital without perturbing the other orbitals
or the phonon state of the molecule. All the details of the interaction are contained
in the quantities C(h̄ω, α, |~k|), which have the dimensions of an energy. In general,
4.3. PHOTOEMISSION: THE SUDDEN APPROXIMATION
43
these coefficients depend on the radiation frequency ω, on the wavevector ~k of the
photoelectron, and on the particular orbital α, where electron is emitted. The
angular integration on all the photoelectron at fixed |~k|, eliminates the dependence
of C on the degenerate component µ within a multiplet.
If we are interested at an energy range corresponding to photoemission from
some precise orbital ᾱ, only one term of Eq. (4.12) is relevant, and the spectrum (4.11) becomes
Ii (E) =
1 XX
†
|hΨf |cᾱµ
|Ψi i|2 δ E − (Ef − Ei )
dᾱ µ f
(4.13)
or in term of the Green’s function (4.8)
Ii (E) = −Im
†
|hΨf |cᾱµ
|Ψi i|2
1 XX
πdᾱ µ f E + i0+ − (Ef − Ei )
(4.14)
where dᾱ is the degeneration of the orbital ᾱ. The choice made in Eq. (4.14) for
the coefficient 2π
|C(h̄ω, ᾱ, |~k|)|2 = d1ᾱ fixes the normalization of the spectrum
h̄
Z
+∞
Ii (E) dE = 1.
(4.15)
−∞
This arbitrary choice is handy for comparison with experimental data, which are
usually only available in “arbitrary units”, i.e. up to a multiplicative constant.
After the sudden appearance of the hole in the molecular orbital ᾱ, µ, the ion
evolves according to the dynamics described by some electron-phonon Hamiltonian,
containing the displaced equilibrium position of the ion with respect to the neutral
species. This dynamical evolution is expressed in the energy domain by the spectral
function Ii (E).
When an electron is ejected from the fivefold degenerate hu HOMO of C60 (the
ᾱ orbital), the dynamics is described by the hu ⊗ (Hg + Gg + Ag ) JT Hamiltonian (3.6). The photoemission spectrum I(E) at finite temperature becomes, using
Eq. (4.3),(4.14)
I(E) =
1 X e−βEi
Iiµ (E)
5 µ
Z
Iiµ (E) = −
|hΨf |c†µ |Ψi i|2
1X
Im
π f
E + i0+ − (Ef − Ei )
(4.16)
(4.17)
where Iiµ is the contribution of the different initial vibrationalP
states i and different
electronic components µ = 0, ±1, ±2 of the HOMO, and Z = i e−βEi accounts for
44
CHAPTER 4. THE PHOTOEMISSION SPECTRUM
the thermal distribution of the initial vibrational states of neutral C60 . In Eq. (4.17)
and in the following we omit the label ᾱ =HOMO, to which we concentrate our
attention.
In this context, Ψi represent the starting vibrational eigenstates of the neutral
C60 . These are unaffected by vibronic interaction and are therefore simple harmonic
eigenstates of the 66 harmonic oscillators in Hvib of Eq. (3.8), of energy Ei =
P
h̄ωΛj (vΛj + 21 ), where vΛj are the quantum vibrational numbers labeling the C60
eigenstates. In Eq. (4.17), c†µ |Ψi i is the initial excitation, right after the electron
has been ejected from orbital labeled µ of the fivefold degenerate hu HOMO, but
the phonon had no time to react yet. Its overlap with each DJT vibronic eigenstate
of C+
60 , Ψf , of energy Ef , gives the intensities of that spectral line.
The calculation of the photoemission spectrum of C60 requires therefore complete
knowledge of the neutral and singly-ionized states. In the next chapter, we describe
the simple but instructive case of the coupling to nondegenerate vibrations, which
can be solved analytically. For JT-coupling to degenerate vibrations relevant for
C+
60 , analytic tools fails, and recursive numerical method, such as Lanczos method,
are called for: to these we devote Chapter 6.
Chapter 5
Nondegenerate vibrational modes
In this chapter, we investigate the e-v coupling of a generic (possibly degenerate)
x state to nondegenerate vibrations, such as the two active Ag modes of C60 . The
results in this chapter are valid independently of the symmetry point group of the
system, and of the symmetry (and degeneracy) of orbital x. For simplicity,, we
label the nondegenerate representation of the vibration as A (as, for example, in
the icosahedral group). The A modes couple to the total charge, and can therefore
be treated separately. The trivial effect of electron-phonon coupling to nondegenerate vibrations is to shift the equilibrium positions of the oscillator. Any possible
electronic degeneracy is not split.
The x ⊗ A Hamiltonian can be treated analytically, and the spectra calculated
exactly. This simple theory is interesting per se, since it applies to all molecules (e.g.
H2 O) whose point group affords only nondegenerate irreducible representations.
We show some photoemission spectra, highlighting their dependence on coupling
and temperature, to understand several general properties which will be useful to
interpret the more complicated JT spectra of degenerate modes.
We will show that each A mode acts independently of all other modes, thus,
in particular, the nondegenerate modes can then be left out of the numerical calculation needed by the degenerate ones, and included analytically in a systematic
way. As a preliminary exercise to apply the derived results, we compute a fictitious
spectrum of C60 , treating all the modes as if they were nondegenerate.
5.1
x ⊗ A - shifted oscillator
Using the notation of Eq. (3.6), the Hamiltonian describing n holes (or electrons)
in a generic orbital x coupled linearly to a nondegenerate molecular vibration can
45
46
CHAPTER 5. NONDEGENERATE VIBRATIONAL MODES
be written has
H=
X
µ
c†µ cµ +
Kgh̄ω X
h̄ω 2
(P + Q2 ) +
Q c†µ cν A Cµ −ν
2
2
µν
(5.1)
Here Q and P are, respectively, the normal coordinate and the canonical coniugate momentum operators of the totally symmetric nondegenerate vibration A of
frequency ω. g is the electron-vibration dimensionless coupling. is the energy of
the electronic orbital and c†µ are the fermion operators creating an hole in orbital
µ with the degenerate manifold. A Cµν are the Clebsch-Gordan coefficients of the
symmetry group: for a d-degenerate electronic state we have A Cµ −ν = √1d δµν . The
√
P
P
†
constant K = d is chosen in such a way that
µν Kcµ cν Cµν =
µ nµ = n,
the number of holes. We see that the nondegenerate vibration A couples only to
the total charge, without breaking the orbital degeneracy. For fixed charge, the
Hamiltonian (5.1) separates in an electronic:
H0 = n
(5.2)
and in vibrational Hamiltonian
HA =
ngh̄ω
h̄ω 2
(P + Q2 ) +
Q
2
2
(5.3)
where the subscript A underlines this is the specific expression for the coupling to
a nondegenerate mode. Eq. (5.3) can be easily simplified
ngh̄ω
h̄ω 2
(P + Q2 ) +
Q=
2
2
i
1
1
h̄ω h 2
2
2
2
P + Q + ng Q + (ng) − (ng) =
=
2
4
4
h
i
h̄ω 2
1 2 1
=
P + Q + ng − (ng)2 .
2
2
4
HA =
Making the canonical shift Q0 = Q + 21 ng, P 0 = P the vibrational Hamiltonian
rewrites as:
h̄ω
h̄ω 02
(ng)2
(5.4)
HA =
P + Q02 −
2
8
The relevant part of the Hamiltonian (5.4), is the familiar harmonic oscillator Hamiltonian. The effect of the addition of n holes (0 → n) in the x orbital is to displace
the equilibrium position to Q = − 21 ng, and shift rigidly the energy levels by a
constant n − h̄ω
(ng)2 , as shown schematically in Fig 5.1.
8
The eigenvalues of (5.4), accounting for the non-trivial shift − h̄ω
(ng)2 are sim8
ply:
1
1
2
(5.5)
Ew = h̄ω (w + ) − (ng)
2
8
5.1. X ⊗ A - SHIFTED OSCILLATOR
47
E
hω
Q
1 (ng)2hω
8
hω
1 ng
2
Figure 5.1: Potential shift due to the coupling to a nondegenerate mode. This trivial electron-phonon coupling does not break the orbital degeneracy. The only effect is
to distort the molecule by an amount − 12 ng, lowering rigidly the energy spectrum by
− 18 (ng)2 h̄ω, without changing the normal frequency.
and the corresponding eigenstates:
ϕw (Q0 ) = ϕw (Q +
1
ng)
2
(5.6)
where ϕv are the harmonic-oscillator eigenfunctions, of Eq (2.38). The total eigenstates then simply factorize
1
ng)
2
(5.7)
1
1
2
= n + h̄ω (w + ) − (ng) .
2
8
(5.8)
Ψfµi ...µn w = ψµ1 ...µn (r1 . . . rn ) ϕw (Q +
and their eigenvalues write
Eµi ...µn w
ψµi ...µn are the wavefunctions describing n holes in the orbital x and ri are the holes
coordinates. The eigenvalues Eµi ...µn w depend only on the total charge, reflecting the
fact that the coupling to a nondegenerate mode does not split the orbital degeneracy.
From now on, we set = 0 to eliminate the trivial electronic contribution n and
focus on the vibrational shift (5.5).
48
CHAPTER 5. NONDEGENERATE VIBRATIONAL MODES
For n = 1, the eigenstates of Eq. (5.6) are just the final eigenstates to which photoemission leads. Then the photoemission spectrum can be calculated analytically
in terms of the overlap of the final and initial states where sudden photoemission
has created a hole. The initial state Ψiv are characterized only by the vibrational
quantum number v and expressed in term of the eigenstates ϕv , with eigenvalues
Ev , of the normal harmonic oscillator in the undistorted configuration, given in
Eqs. (2.38),(2.37). Throughout this chapter, we label with v the vibrational quantum number of the initial state (2.38),(2.37), and with w the final ones (5.6),(5.5).
The overlap between the initial and final state, factorized in the electronic and
vibrational contribution:
hΨfν w |c†µ |Ψiv i = hψν |ψµ ihϕw |T α |ϕv i = δνµ hϕw |T α |ϕv i
(5.9)
where T α is the α-translation of the normal coordinate Q with α = 21 ng. (T α )† |ϕw i
is the final vibrational eigenstates of Eq. (5.6), and ψµ is the hole wavefunction.
FromP
now on, we make abstraction of the electronic part of the matrix element,
1
as d µ δµµ = 1. Then, for an initial state v, the poles and residues of Gvv (E)
(Eq. (4.5)) are
1
Ewv = Ew − Ev = h̄ω w − v − (ng)2
8
Z
2
∞
v
Rw
= |hϕw |T α |ϕv i|2 = ϕw (Q + α)ϕv (Q)dQ
−∞
(5.10)
(5.11)
The residues can be calculated analytically, using the explicit expression for ϕv/w
in term of the Hermite polynomials given in Eq. (2.38):
Z ∞
α
hϕw |T |ϕv i =
ϕw (z + α)ϕv (z)dz =
−∞
Z ∞
(z+α)2
z2
= N v Nw
e− 2 e− 2 Hw (z + α)Hv (z)dz =
−∞
Z ∞
2
α 2
− α4
e−(z+ 2 ) Hw (z + α)Hv (z)dz =
= N v Nw e
Z−∞
∞
α2
α
α
2
= N v N w e− 4
e−z Hv (z − )Hw (z + )dz =
2
2
−∞
2
√
α2
α w−v w−v α
Lv
w≥v
= Nv Nw e− 4 2w π v!
2
2
where the last equality is a known result in the mathematics of special functions [11].
Lλr (z) are the Laguerre polynomials of degree r
z −λ ez dr r+λ −z λ
Lr (z) =
z e
,
(5.12)
r! dz r
5.1. X ⊗ A - SHIFTED OSCILLATOR
49
and Nv/w is the normalization constant of the harmonic oscillator wavefunction
defined in Eq. (2.38). Substituting its explicit expression, and with some simplifications we get the final result
r
2v v! − α2 w−v w−v α2 e 4 α
hϕw |T α |ϕv i =
Lv
.
(5.13)
2w w!
2
which holds even for w < v.
Practically, only a finite number of state can be considered, as for large w the
overlap (5.11) with the initial states goes to 0, as shown by the asymptotic limit of
Eq. (5.13), for w v
hϕw |T α |ϕv i '
wv
α
√
2
√
w−v
w! v!
,
[w v].
(5.14)
In particular, the number of nonnegligible overlaps (5.13) depends both on the
distortion α = g/2,√and on the initial state v considered. For strong coupling the
diverging term (α/ 2)w−v contrasts the factorial w!, slowing down the approach
to 0. The same consideration applies to the w v factor, for large values of v. This
means that for low initial excitation and weak coupling relatively few states have
significant residues, while for large α and/or v a lot of residues are comparable. In
the special case v = 0 (accounting for the T = 0 spectrum), the Laguerre polynomial
is unity and the matrix element simplifies to
w
α
1 − α2
α
√
hϕw |T |ϕ0 i = √ e 4
(5.15)
2
w!
This has an asymmetric bell shape as a function of w, with the maximum close to
(α2 − 1)/2.
Given the analytic expression for poles (5.10) and residues (5.11), the photoemission spectrum for initial state is expressed, in term of the Green’s function (4.14),
as
v
Rw
1
1X
Im
(5.16)
Iv (E) = − Im Gii (E + i0+ ) = −
π
π w
E + i0+ − Ewv
v
with Ewv and Rw
given by Eqs. (5.10),(5.11).
Before investigating thermal effects, it is instructive to study the shape of spectra
produced by a given initial state. In Fig. 5.2, we compare the spectra from initial
states v = 0 (ground state), 1, 3 and 10 phonons, for the following values of coupling:
g = 0.1, 1, 2, 3, 5, 10. For small g the e-v interaction induces only a weak change in
the eigenfunction, and few weak satellite structures appear in the spectra, regularly
separated by h̄ω. In particular, for g = 0.1 no visible structures appear, meaning
50
CHAPTER 5. NONDEGENERATE VIBRATIONAL MODES
g=0.1
-5
-4
-3
-2
-1
0
1
2
3
4
g=1
5
-5
-4
-3
-2
E [hω]
-1
0
1
2
-6
-4
-2
0
E [hω]
4
5
-8
-6
-4
-2
E [hω]
2
4
6
8
0
2
4
g=5
10 -15
-10
-5
0
5
E [hω]
6
8
E [hω]
g=3
-10 -8
3
g=2
10
15
g=10
20 -30
-20
-10
0
10
20
30
E [hω]
Figure 5.2: Comparison of the spectra for different values of coupling g = 0.1, 1, 2, 3, 5, 10.
Each frame contains, from top to bottom, the photoemission spectra for initial excitation
characterized by v = 0, 1, 3, and 10 phonons. The vertical dashed line marks the final
ground-state energy position −g 2 h̄ω/8. The sticks are normalized so that the sum of
the intensities equals unity. The vertical scale is the same for all panels within the same
frame.
5.1. X ⊗ A - SHIFTED OSCILLATOR
51
that the initial states are substantially unperturbed by the e-v coupling. This is
reflected also by the imperceptible ground state energy lowering (vertical dashed
line in Fig. 5.2). Increasing the coupling, the spectral weight displaces to final states
corresponding to different phonon numbers. Starting from the ground state, only
higher energy peaks appears, while for excited initial states also some lower-energy
anti-Stokes structures appear, corresponding to less phonons in the final states. As
is especially clear for strong coupling, we find an anti-Stokes peak for each of the
accessible states, i.e. one peak for v = 1, corresponding to a transition to the ground
state w = 0 of the distorted oscillator; three peaks for v = 2 corresponding to state
with w = 0, 1 and 2 phonons and so on. For very strong coupling, in particular
at g = 10, the final ground-state eigenfunction is substantially orthogonal to the
initial one, due to the large distortion. Here the spectral weight is spread on a lot
of final states. Note the approximate symmetry of these spectra around the zeroenergy position, reflecting the fact that the final and initial mean energy, remains
approximatively the same.
For a system in thermal equilibrium, a statistical approach is necessary, and the
characteristic effects of temperature on the spectra are readily investigated through
this simple model. The spectrum at finite T given by Eq. (4.9) can be written as
I(E) = −
where Z =
P
v
v
1X
2 X e−βEv
Rw
(T )
Iv (E) = −
Im
+
h̄ v
Z
π vw
E + i0 − Ewv
(5.17)
e−βEv , and
v
Rw
(T ) =
1 −βEv v
e
Rw ,
Z
(5.18)
v
with Ewv and Rw
are given by Eqs. (5.10),(5.11). The sum on all initial final states
v, w amounts to a sum of all poles, with suitable weights.
In Fig. 5.3, we shown a few spectra, calculated at different temperature: T =
0.2 h̄ω, 4 h̄ω, 20 h̄ω. Because of the Boltzmann factor e−βEv , at low temperature
substantially only the ground state is involved, and the spectra almost coincide
with the 0-phonons spectra of Fig 5.2. At higher temperature, the main effect is to
broaden the spectra, eventually reaching a Gaussian-like structure centered at E =
0. The broadening effect is stronger at larger coupling. At g = 0.1 only exceedingly
weak satellites appear, even at a relatively high temperature (T = 20h̄ω). For large
coupling the contribution of many excited states becomes important and the large
broadening makes the spectra very flat.
For an A mode, the overall effect of temperature is to make the spectra broaden
and more regular, but not any denser: the constant separation h̄ω of neighboring
structures is maintained in temperature, due to Eq. (5.10).
52
CHAPTER 5. NONDEGENERATE VIBRATIONAL MODES
g=0.1
-30
-20
-10
0
10
20
g=1
30 -30
-20
-10
E [hω]
0
10
-20
-10
0
E [hω]
30 -30
-20
-10
E [hω]
10
20
0
10
g=5
30 -30
-20
-10
0
E [hω]
20
30
E [hω]
g=3
-30
20
g=2
10
20
g=10
30 -30
-20
-10
0
10
20
30
E [hω]
Figure 5.3: Temperature dependence of the photoemission spectrum, for several values of
the coupling g. From top to bottom, each frame contains the spectrum corresponding to
kB T = 0.2 h̄ω, 4 h̄ω, 20 h̄ω. The vertical dashed line marks the final ground-state energy
position −g 2 h̄ω/8.
5.2. THE A MODES IN A MANY-MODE CONTEXT
5.2
53
The A modes in a many-mode context
In this Section we extend the results of the previous Section to many-mode systems,
in which, also degenerate modes may be active.
In the case of a degenerate orbital x coupled to degenerate vibrations X, the
coupling breaks the orbital degeneracy. The interaction of a given degenerate mode
with the hole is correlated to the dynamics of all other modes interacting with the
same hole state at the same time. They cannot be separated from each other, and no
exact relation of the type (5.10),(5.11) are applicable for the degenerate JT modes.
This more involved situation requires a numerical treatment introduced in Chapt. 6.
However, it is still exactly true, regardless of any possible splitting of the electronic
degeneracy, that the totally symmetric non degenerate A vibrations only couple to
the total charge in the orbital, and remain therefore completely decoupled from each
other and from all modes of different symmetry X. The analytic treatment of A
modes of previous Section, remains therefore relevant to the many-mode context.
The Hamiltonian of a many-mode system is conveniently separated into two
contribution:
X
H=
HiA + H X .
(5.19)
i
The first term contains the HiA harmonic and electron-phonon Hamiltonians of
the nondegenerate A modes, each given by Eq. (5.3) with its specific frequency ωi
and couplings gi ). The latter term H X describes collectively the coupling to all
degenerate modes X and include also the electronic Hamiltonian H0 of Eq. (5.2).
X
We indicate with Ψw̄
(r, QX ) the eigenstates of the Hamiltonian H X with eigenvalues Ew̄X , describing the dynamics of one hole, whose coordinates are r, and of
the nondegenerate normal vibration QX . They are, in general, vibronic states and
do not factorize into electronic and vibrational wavefunctions, as in Eq. (5.7). w̄
globally indicates the quantum numbers characterizing these vibronic states. Then,
the eigenstates of the total Hamiltonian (5.19) are factorized as:
1
1
f
X
Ψw̄
w1 w2 ... (r, Qw̄ , Q1 , Q2 . . .) = Ψw̄ (r, QX ) ϕw1 (Q1 + ng1 ) ϕw2 (Q2 + ng2 ) . . . (5.20)
2
2
and the corresponding eigenvalues are
Ew̄ w1 w2 ... = Ew̄X + Ew1 + Ew2 + . . .
(5.21)
ϕwi and Ewi are the eigenstates of the Hamiltonian HiA , for each of the nondegenerate modes, given by Eqs. (5.6),(5.5), with coordinates Qi .
As in Eq. (5.9), the overlap between initial and final states factorizes
X †
hΨfν w |c†µ |Ψiv i = hΨw̄
|cµ |Ψv̄X ihϕw1 |T α1 |ϕv1 ihϕw1 |T α1 |ϕv1 i . . .
(5.22)
54
CHAPTER 5. NONDEGENERATE VIBRATIONAL MODES
The initial states are given in terms of the wavefunctions Ψv̄X , of energy Ev̄X , describing globally the degenerates modes X and the electronic motion within the
degenerate shell, and of the vibrational eigenstates ϕvi of each harmonic oscillator
of symmetry A, whose eigenvalues are Evi . hϕwi |T αi |ϕvi i are again the residues Rvwii
given by Eq. (5.11) for a single nondegenerate mode. Then the poles and residues
of the x ⊗ (X + A1 + A2 . . .) system can be written compactly as:
Ew̄v̄ vw11vw2 ...
=
2 ...
v̄ v1 v2 ...
Rw̄
w1 w2 ... =
X v̄
Ew̄
X v̄
Rw̄
+ Ewv11 + Ewv22 + . . .
v1
v2
Rw
Rw
...
1
2
(5.23)
(5.24)
vi
in terms of the poles Ewvii and residues Rw
of each A mode, which can be calcui
lated analytically through Eqs. (5.10),(5.11),Pand those of the degenerate modes
X †
X 2
v̄
X, given by XEw̄v̄ = Ew̄X − Ev̄X and XRw̄
= d1
µ |hΨw̄ |cµ |Ψv̄ i| , which include the
v̄
sum on the orbital components µ. XEw̄v̄ and XRw̄
do not have analytic expressions
like (5.10),(5.11) and necessitate a numerical approach. In particular for nontrivial
JT-systems the poles XEw̄v̄ are not simply multiples of the original frequency as in
Eq. (5.10).
The total spectrum is then written, using (5.23) and (5.24), as
I(E) =
X
v̄ v1 v2 ...
1
Iv̄ v1 v2 ... (E) = −
π
1
X
e−β(Ev̄ +Ev1 +Ev2 ...) Iv̄ v1 v2 ... (E)
ZX Z1 Z2 . . .
X
w̄ w1 w2 ...
v̄ v1 v2 ...
Rw̄
w1 w2 ...
Im
+
E + i0 − Ew̄v̄ vw11vw2 ...
2 ...
(5.25)
(5.26)
Using (5.23),(5.24),(5.18) we can write the spectrum at finite temperature, in a
more useful form similar to Eq. (5.17), as
1
I(E) = −
π
X
v̄ v1 v2 ...
w̄ w1 w2 ...
E
v2
X v̄
v1
(T ) . . .
(T ) Rw
Rw̄ (T ) Rw
2
1
v1
v̄
+
X
+ i0 − ( Ew̄ + Ew1 + Ewv22 +
. . .)
(5.27)
in terms of the thermally weighted residues of each separated subsystem, introduced
in Eqs. (5.18).
The important result obtained here is that we can approach the complicated
problem of the coupling to degenerate modes, completely independently from nondegenerate ones, whose contributions can be calculated analytically and included
through Eqs (5.23),(5.24),(5.27). This reduces the number of active degrees of
freedoms, making the numerical approach, required by the really JT-coupled degenerates modes, easier.
In the next Section we apply these analytical results to simple case of two A
modes, to show the effect of the coupling to many modes in the most transparent
5.3. TWO A MODES
55
situation. Some features of the spectra obtained, are characteristic of the many
modes system, and their investigation are a preliminary step to the understanding
of more complicated systems, such as C60 . We close the chapter, computing a
fictitious spectrum of C60 , treating all the modes as they were nondegenerate.
5.3
Two A modes
The general results of Sect. 5.2 apply, in particular, to the the case of many nondegenerate A modes coupled to same orbital x. The spectra for such a system can
be computed analytically through Eqs. (5.27),(5.18) and the expressions for the
poles (5.10) and residues (5.11).
In this Section we study the system of two A modes, A1 and A2, of frequency
respectively h̄ω and 6.2 h̄ω. In Fig. 5.4 we show the spectra for different values of
the dimensionless coupling g1 = 1, 2, 3, 5, at a fixed value of the coupling g2 = 1, at
a temperature kT = 2 h̄ω. We choose these values looking at the problem of C60
which is the object of this work. In fact, the ratio between the modes A2 and A1
mimics the ratio between the highest and lowest frequency modes of C60 , Hg 8, of
coupling g ≈ 1, and Hg 1 of coupling g ≈ 3 (Table 3.1). They are also the most
strongly coupled and we will show them to be responsible of the important features
of the C60 photoemission spectrum. Also, for mode Hg 1 the temperature of 2 h̄ω
corresponds at about the 800 K of the experimental measurement.
We plot the spectra A1+A2 of this two modes system, together with those of the
individual single modes, suitably shifted to align the ground-state energy. In fact,
2
each mode gives independently an energy shift of − 81 g1/2
as show schematically in
Fig. 5.1. We broaden the spectra with Lorenzians of width = 0.5, and show them
together with their “stick” original structures. For relatively weak coupling of A1,
g1 = 1 (left top of Fig. 5.4), the gross structures of the spectrum are determined by
the higher frequency mode A2: we find a satellite exactly 6.2 h̄ω above the main
peak, given by the A2 1-phonon excitations. Also, a weak 2-phonons structure
appears at 12.4 h̄ω, but it is flattened by the adding of the low-frequency mode
A1. The effect of the low-frequency mode A1 is just to broaden the peaks, through
regular minor structures appearing at exactly h̄ω separations. At larger coupling,
g1 = 2 (right top), the broadening of A1 increases. The important features are still
those of the A2 mode, but the A1 broadening displaces some spectral weight away
from the main structures and the satellite at 6.2 h̄ω becomes broader. For g = 3 (left
bottom) the contribution of the A2 mode appears only as an asymmetric shoulder
in an otherwise mainly A1 spectrum. For a coupling as large as g = 5 (right
bottom), we recover a Gaussian-like shape, while the 6.2 h̄ω satellite completely
disappears. Looking at the stick structures of the spectra we can still see the weak
poles corresponding to 1-A2-excitations at (6.2 ± n) h̄ω, aside of those of 0-A2-
56
CHAPTER 5. NONDEGENERATE VIBRATIONAL MODES
g1=1
-10
-5
g1=3
-10
-5
A1
A2
A1+A2
0
5
E [hω]
g1=2
10
15 -10
A1
A2
A1+A2
0
5
E [hω]
-5
g1=5
10
15 -10
-5
A1
A2
A1+A2
0
5
E [hω]
10
15
10
15
A1
A2
A1+A2
0
5
E [hω]
Figure 5.4: Solid lines: photoemission spectra for two nondegenerate modes A1 and A2,
of frequency respectively h̄ω and 6.2 h̄ω, for different values of the dimensionless coupling
g1 = 1, 2, 3, 5, for g2 = 1. The single-mode spectra (dashed) are shown for reference
and shifted to the same ground-state energy of the two-mode calculation. The main
satellite corresponds to the characteristic 1-A 2 -phonon excitation, at exactly 6.2 h̄ω from
the main peak. The lower-frequency mode, with its dense structures at h̄ω, accounts for
the broadening of both the main peak and the satellite and at strong coupling flattens
them completely.
5.4. 66 A MODES: A FICTITIOUS SPECTRUM OF C60
57
phonons at ±m h̄ω, but the broadening covers easily these details.
In Fig. 5.5 we show the same calculation for larger values of the A2 coupling,
g2 = 3. The difference from Fig. 5.4 is that several strong structures appear,
corresponding to many-phonons excitations of the higher-frequency mode. But the
interpretation of the spectral shape remains the same: the higher-frequency modes
A2 characterizes the main features of the spectrum, while the lower-frequency one,
A1, more sensible to thermal effects, accounts for the observed broadening. Note
that the overall broadening of the lower right panel, where both g1 and g2 are large,
follows the envelope of the A2 spectrum.
These conclusion are bound to apply to the photoemission spectrum of any
many-modes system characterized by some low frequency modes (h̄ω < kB T ) which
are strongly thermally excited, plus some others high frequency mode (h̄ω > kB T ),
whose contribution is to construct the gross structures of the spectrum. We are
likely to find the same roles of different classes of modes even in the JT spectra.
However, because the electronic and vibrational excitations are mutually entangled
in the JT final state coupling, the vibronic excitation are bound to lose the regular
phonon-like fine structure characteristic of A modes and well illustrated in Fig. 5.4
and 5.5.
5.4
66 A modes: a fictitious spectrum of C60
To conclude the chapter we apply the results of the previous Sections to calculate
analytically the spectrum of a fictitious C60 , with all 66 active modes treated as
nondegenerate A modes. We are principally interested to highlight, through the
comparison with experimental spectra [1, 2], the features which are not explainable by this approach, and which are instead typical JT-features associated to the
degeneracy of the active modes.
For its calculation we use the frequencies of Table 3.1, and such coupling as to
give the same adiabatic energy gain of the degenerate modes, also accounting for
the zero-point quantum correction. For degenerate modes the adiabatic energy gain
is given by a different expression from that of the nondegenerate modes. However
it still holds true that the simultaneous coupling of several modes will generally
lead to a cumulative classic distortion and to an adiabatic energy gain which is the
sum of the individual energy gains [45, 46]. Given the energy gain for the A mode
EA = −(1/8)gA2 , for the H modes EHg = (1/18) (gHg sin α)2 + (1/10) (gHg cos α)2
2
and for the G modes EGg = (1/18) gG
we use the relation
g
E adi 1
1 2
g̃Λ = 0
EΛ .
8
E0 d Λ
(5.28)
With g̃ we indicate effective couplings for each A mode representing a mode of
58
CHAPTER 5. NONDEGENERATE VIBRATIONAL MODES
g1=1
-20
-10
g1=3
-20
-10
A1
A2
A1+A2
0
10
E [hω]
g1=2
20
-20
A1
A2
A1+A2
0
10
E [hω]
-10
g1=5
20
-20
-10
A1
A2
A1+A2
0
10
E [hω]
20
A1
A2
A1+A2
0
10
E [hω]
20
Figure 5.5: Solid line: Photoemission spectra for a system of two nondegenerate modes
A1 and A2, of frequency respectively h̄ω and 6.2 h̄ω, for different values of the dimensionless coupling g1 = 1, 2, 3, 5, for g2 = 3. The single-mode spectra (dashed) are shown
for reference and shifted to the same ground-state energy of the two-mode calculation.
The important features of the spectrum correspond to the characteristic higher-frequency
phonons excitations, at 6.2 h̄ω separation. The lower-frequency mode, with its dense
structures at h̄ω, accounts for the broadening of the A 2 peaks and at strong coupling
completely washes them out.
5.4. 66 A MODES: A FICTITIOUS SPECTRUM OF C60
59
Intensity [arb. units]
analytical calculation
experiment
0
Binding energy [meV]
Figure 5.6: Fictitious C60 spectrum, calculated treating all the 66 vibrational normal
modes as nondegenerate A modes, rescaling the actual couplings of C 60 (Table 3.1) to
give the correct JT energy shift. Comparison with the measured spectrum [1] (The
broadening = 10 meV mimicks the declared experimental resolution [1]).
symmetry Λ. The factor 1/dΛ accounts for the fact we take dΛ A modes for each
dΛ -degenerate mode Λ. The remaining factor is the ratio between the zero-point
adiabatic energy E0adi and the exact quantum value E0 computed in Chap. 7. We
rescale all modes to obtain the correct energy gain. We chose to ignore the fact that
Gg modes and the sin α part of the Hg modes contribute to the D3 minima, instead
of the D5 , as these are small it would not make much difference even to leave them
out.
In Fig. 5.6 we show the spectrum corresponding to the system of these 66 nondegenerate modes, together with the measured spectrum [1], shifting the spectra
(i.e. the HOMO position with respect to the vacuum) to have them superimposed.
We see that the general amount of broadening and the asymmetric shape of the
spectra are accounted for by this simple model. But important features are missed
by this calculation:
a) the main satellite at about 230 meV from the main structure disappears completely. This peak found at a frequency higher than each of the characteristic frequency: it must reflect some typical DJT effect as shell investigate in
Chap. 7.
b) the secondary structures at multiples of h̄ωHg 1 in the calculated spectra miss
60
CHAPTER 5. NONDEGENERATE VIBRATIONAL MODES
the smooth structureless shape of the measured one. This reflects the fact
that degenerate modes do not produce simple vibrational levels in the final
state, but vibronic ones, strongly coupling dependent.
Chapter 6
The degenerate modes: Lanczos
diagonalization
A degenerate molecular orbital coupled to degenerate vibrations, splits the orbital
degeneracy. The motions of the degenerate modes are strongly correlated and cannot be separated from each other, as could be done for A modes. No exact relation
of the type (5.10,5.11) are applicable here, and numerical approaches are needed,
due to the large size of the Hilbert space.
In short, the problem we have to face is the diagonalization of the Hamiltonian
matrix (3.6), or equivalently the resolvent (4.4), calculated on harmonic oscillator
basis, in principle infinite dimensional. In practical only relatively few final states, in
the distorted configuration, have nonnegligible overlaps with the initial excitation,
as seen for A modes in Sect. 5.1 and thus a finite number of orthonormal basis states
can be considered. However, to obtain converged results, the basis to consider must
often contain several millions states, due to the relative complexity of C+
60 ion (it has
66 coupled vibrational degrees of freedom, plus the hole dynamics). Moreover finite
temperature calculations imply numerous initial excitations, making the numerical
approach computationally heavier.
The molecular symmetry and the form of the Hamiltonian is helpful in that, for
a suitable choice of the basis, only a small fraction of the matrix elements of the
linear JT-Hamiltonian (3.6) is nonzero. Then H can be represented by a sparse
matrix with N st rows and a few (about 7) nonzero elements in each row. This way,
the memory required for storing the Hamiltonian matrix are not very severe, and
matrices of N st ≥ 106 are treated routinely within computer memories of about
1 GB. Finding eigenvalues and eigenvector of such large matrices is not possible
with standard algorithm performing the full diagonalization. The separation of the
nondegenerate A modes from the numerical calculation, illustrated in the previous
Section, and symmetry considerations improve this situation only marginally. To
approach the problem, one must instead resort to the power algorithms, among
61
62
CHAPTER 6. DEGENERATE MODES
which the Lanczos algorithm [47] is one of the most widely known and most efficient.
In this Chapter we introduce the Lanczos method, and its specific application to
the calculation of photoemission spectrum from linear JT molecules, in particular
from C60 . We test the method through the calculation of single-mode, and few
modes spectra at T = 0 and in Sect. 6.3 we show the results of the calculation
of the collective contribution of the nondegenerate modes to the C60 spectrum at
T = 0, concentrating on its convergence. We devote the next Chapter to the
comparison with the measured spectrum, and its interpretation.
In spite of the impossibility of the calculation of C60 spectrum at high temperature (T > 100 K), we finally describe the specific technique used to approach
finite temperature, based on the random sampling of the ensemble of the initial
thermal excitations. We show the results of single-mode calculations, in particular
for the lowest frequency modes Hg 1, which together with the T = 0 total spectrum
of C60 , will be the ingredient that, in the next Chapter, allows us to interpret the
experimental spectra.
6.1
The Lanczos algorithm
The Lanczos method is a very convenient approach for the determination of eigensolutions of matrices, especially those of very large size and sparse character (i.e.
with many matrix elements equal zero). The essential principle of the Lanczos
recursion method is very simple and very general at the same time. Consider a
quantum system, a Hermitian operator H, and a number N st (arbitrary large) of
orthonormal basis states {Ψk } (k = 1, 2, . . . , N st ). Starting from any given state
|f0 i belonging to the space spanned by {Ψk }, and operating with H, the recursion
method provides a one dimensional chain representation of the original quantum
system.
To illustrate the procedure, consider an operator H (usually the Hamiltonian of
the system) and an initial normalized state f0 , arbitrarily chosen; the starting state
f0 is often addressed as the seed state of the procedure. We apply the operator H
to the initial state and subtract from H|f0 i its projection on the initial state, so to
obtain a new state F1 , orthogonal to f0 . The operation of orthogonalization can be
conveniently performed by means of the projection operator P0 = |f0 ihf0 |, and the
state F1 can be expressed as
|F1 i = (1 − P0 ) H|f0 i = H|f0 i − a0 |f0 i
(6.1)
a0 = hf0 |H|f0 i.
(6.2)
where clearly
Let us indicate with b1 the norma of F1 , and with f1 the corresponding normalized
63
6.1. THE LANCZOS ALGORITHM
state, namely
1
|F1 i.
(6.3)
b1
In a rather similar way we proceed now by applying the operator H to the state
|f1 i; then H|f1 i is orthogonalized to both |f1 i and |f0 i, obtaining the state |F2 i.
We have
|F2 i = (1 − P1 ) (1 − P0 ) H|f1 i = (1 − P1 − P0 ) H|f1 i
(6.4)
b21 = hF1 |F1 i,
|f1 i =
where P1 = |f1 ihf1 | denotes the projection operator on the state f1 . We thus obtain
where
|F2 i = H|f1 i − a1 |f1 i − b1 |f0 i
(6.5)
1
|F2 i.
(6.6)
b2
and the last term b1 |f0 i in Eq. (6.5), is a direct consequence of (6.1) and (6.3).
Let us now proceed to the next logical step of the sequence, and here we arrive
at the key point of the ingenious, and yet so simple, Lanczos procedure. Consider
in fact the state F3 obtained by orthogonalization of H|f2 i to the previous states
f0 , f1 , f2 ; we have
a1 = hf1 |H|f1 i,
b22 = hF2 |F2 i,
|f2 i =
|F3 i = (1 − P2 )(1 − P1 )(1 − P0 )H|f2 i = (1 − P2 − P1 − P0 )H|f2 i
(6.7)
We notice that P0 H|f2 i ≡ 0 since hf0 |H|f2 i ≡ 0: thus the iterative procedure
leads to a three-terms relation. This is the most remarkable feature of the Lanczos
method: that it provides a recursive relation based on three terms only.
Iterating n times this procedure, orthogonalizing the new generated state H|fn i
only with respect to its two predecessors |fn i and |fn−1 i, we obtain
|Fn+1 i = H|fn i − an |fn i − bn |fn−1 i.
(6.8)
The next pair of coefficients are given by
b2n+1 = hFn+1 |Fn+1 i,
an+1 = hfn+1 |H|fn+1 i.
(6.9)
The three term relation (6.8) is all we need to generate the orthonormal basis
|f0 i, |f1 i, . . . and the set of coefficients {an } and {bn }. Having to deal with only
three large vector at the same time is a remarkable advantage, both conceptual and
technical, in the specific calculations carried out by a computer code. On the basis
set {|fn i} the operator H is represented by the tridiagonal matrix


a 0 b1
 b1 a 1 b2




b2 a 2 b3
H =
(6.10)




· · ·
· ·
64
CHAPTER 6. DEGENERATE MODES
The operator (6.10) can also be written in the form
H=
∞
X
n=0
an |fn ihfn | +
∞
X
n=0
bn+1 [|fn ihfn+1 | + |fn+1 ihfn |]
(6.11)
The recursion method allows (at least in principle) to bring in a tridiagonal
form the Hamiltonian of any system, with an arbitrary number N st of states. No
explicit diagonalization of the original matrix H is required for performing such a
transformation. The one-dimensional chain representation is interesting first of all
since it describes economically the system through the 2N st − 1 parameters an and
bn , rather than the original N st (N st + 1)/2 matrix elements, for a generic basis (for
N st = 106 for instance, we have packed the spectral information in approximately
two million numbers instead of 5 × 1011 numbers).
An even more crucial feature of the tridiagonal representation (6.10), making the
Lanczos procedure extremely convenient for spectral calculations, is the possibility
to obtain very easily the Green’s function (4.4). In particular, by explicit expansion
of the inverse matrix element G00 in terms of determinants, it is straightforward to
show that the matrix element G00 on the seed state f0 is given by the continued
fraction expansion
G00 (E) = hf0 |
1
1
|f0 i =
E−H
E − a0 −
b21
(6.12)
b2
E−a1 − E−a2 −...
2
It has been verified that in most cases few terms M ∼ 300 are normally sufficient
to get converged spectra over the whole range of energies [48].
Besides computing the Green’s function, the Lanczos method is suitable for
the calculation of approximate ground and excited energies. The first M << N st
steps of the Lanczos chain generate a relatively small tridiagonal matrix HM (6.10),
that is easily diagonalized using standard numerical routines to obtain approximate
eigenvalues j , with corresponding eigenvectors ΨLj , in the full space
|ΨLj i
=
M
X
i=0
vji |fi i,
j = 0...M
(6.13)
given in terms of the eigenvector vji of the tridiagonal matrix HM in the space
spanned by the Lanczos vectors |f0 i, |f1 i . . . |fM i. It is important to realize that
|Ψj i are in general not exact eigenfunction of H, but show a remainder
H|ΨLj i − j |ΨLj i = bM +1 vjM |fM +1 i
(6.14)
Generally, relatively few iterations M are needed to converge the most widely separated eigenvalues, especially those whose eigenfunctions have large overlap with
6.2. THE JT MODES OF C60 : LANCZOS SPECTRA
65
the seed state. Few hundred iterations are typically suitable to our aim [48], and
all the calculated spectra shown in the present work are completely converged with
the M = 300 iterations we have used.
If in equation (6.8) bM +1 happens to vanish, we have found a (M +1)-dimensional
eigenspace where HM is already an exact representation of H. This unavoidably
occurs when M = N st − 1, but for M < N st − 1 it can only occur if the starting
vector f0 is orthogonal to some invariant subspace of H. This is clearly true if the
seed state f0 belong to some particular symmetry species. Symmetry consideration
are therefore very important to improve the numerical calculation.
The first column of matrix vji , containing the projections of the Lanczos eigenvectors |ΨLj i on the basis {|f0 i, |f1 i . . . |fM i}, contains the overlaps vj0 = hf0 |ΨLj i of
the initial state to the eigenstates, which are useful for version (4.11) of the theoretical spectrum. This observation permits us to refrain from ever computing |ΨLj i,
with a large computational advantage.
6.2
The JT modes of C60: Lanczos spectra
In this Section we come to the specific application of the Lanczos method to the
calculation of the photoemission spectrum of C60 , described in term of linear JT
Hamiltonian (3.6) for the final ion C+
60 . Unlike the nondegenerate A modes, whose
dynamical response separates out, degenerate modes Hg and Gg are strongly correlated and cooperate to the final JT vibronic structure. In the medium-coupling
regime of C60 (Table 3.1), neither a static JT treatment, where tunneling between
equivalent minima is neglected, nor the perturbation-theory approach do work.
Therefore we compute the photoemission spectrum by numeric Lanczos diagonalization of the Hamiltonian (3.6), on the product basis of the five HOMO hole
states times the N osc = 64 degenerate Hg and Gg modes of Table 3.1. As the
JT-interaction term (3.9) connects only states with one phonon of difference, on
this basis the Hamiltonian is a very sparse matrix, particularly suitable to Lanczos
method. Moreover symmetry considerations allow us to restrict the calculation to
invariant subspace, considerably relaxing the memory requirement. In fact, states
belonging to different symmetry representations, constitute orthogonal invariant
subspace of H, which are never connected by Lanczos iterations. Starting with a
seed state of given symmetry, we can restrict to the corresponding invariant subspace, cutting off from the basis the eigenstates of different symmetry.
In particular for C60 , each state (6.16) belongs to a C5 representation, labeled
by
!
X
M = µ+
(6.15)
mvΛjm ,
Λjm
66
CHAPTER 6. DEGENERATE MODES
reported into the −2 . . . 2 range by suitable addition of an integer multiple of 5. This
defines five invariant subspaces, that we treat separately, with an improvement of
a factor 5 in the basis size1 .
The states are then conveniently labeled by M plus the vibrational quantum
numbers vΛj
|M ; vHg 1 , . . . , vHg 8 , vGg 1 , . . . , vGg 6 i
(6.16)
where vΛj indicates collectively the set of quantum numbers vΛjm of the degenerate
components, for each of the j modes of symmetry Λ (j = 0 . . . 8 for Λ = Hg
and j = 6 for Λ = Gg ), accordingly to notation of Sect. 3.5. Together with M ,
they identify uniquely the hole state µ through Eq. (6.15), and the sum over µ in
Eq. (4.16) is equivalently replaced by a sum over M .
Moreover, the terms of that sum are not all independent. In fact, M = ±1 and
±2 pairs of C5 representations are complex conjugate representations and produce
therefore the same spectrum. In addition, the kaleidoscope symmetry [49] of the Ih
group guarantees that M = 1 and M = 2 also give equivalent spectra. Therefore,
in practice the only different spectra to be computed are M = 0 and, for example,
M = 1, with a computational time gain of 2/5.
At this point, each invariant subspace corresponding to a fixed M still contains
infinite states, because of the infinite possible bosonic vibrational excitations, thus
some truncation is necessary. We cut off the basis by taking all the states below some
maximum number of quanta vΛj for each mode and dropping the infinitely many
other modes. In fact, as seen in Sect. 5.1 for the trivial case of a single nondegenerate
A mode, the overlaps between an harmonic eigenstate and the eigenstates of a
shifted oscillator rapidly vanish, when quantum number differences become very
large. Each cut basis is then characterized by the set {v̄Λj } of the maximum number
of phonons needed by each degenerate oscillator. To better illustrate the meaning of
this mask, the simple example of Fig. 6.1 is illuminating. We build the basis states,
distributing up to 1, 2, . . . n phonons only among the modes for which n ≤ v̄Λj . For
the example of Fig. 6.1, of a system of three modes and a mask {5, 2, 3}, this means
to distribute up to 2 phonons among all modes, 3 phonons only between the modes
1 and 3, and up to 5 phonons only in mode 1, as represented graphically in the
left side of Fig. 6.1. On the right side we show a more typical truncation, where
a global cutoff is used, with vmax representing the maximum number of quanta we
distribute in all the modes.
It is evident the convenience of the our truncation (left side): setting the cutoff
v̄Λj to a large value for the strongly coupled modes, we include much fewer states,
only those giving the important structures of the spectra. The contribution of the
1
Even larger size gains would be obtained by labeling states with the full icosahedral symmetry,
but that would mean much more intricate basis states: here we prefer to stick to the simple C 5
labeling only
67
6.2. THE JT MODES OF C60 : LANCZOS SPECTRA
v3
v3
vmax
v3
v1
v1
v1
vmax
v2
v2
v2
vmax
Figure 6.1: Left: Schematic description of the basis cut, corresponding to the mask
{v̄1 = 5, v̄2 = 2, v̄3 = 3} for a system of three modes. Each point of the frame is a state,
whose coordinates (v1 , v2 , v3 ) represent the number of quanta in the corresponding mode.
All the states with 2 phonons are retained (pyramidal volume under the plane cutting
the 3 axis). Among the 3-phonons states only those with phonons of species 1 or 3 are
included (trapezoidal areas in the plane 1-3). Finally, states up to five phonons of species
1 are taken (segment from 3 to 5 on axis 1. Right) Standard truncation corresponding to
fix the maximum numbers of quanta vmax we can distribute in all the mode.
dropped states amounts to some broadening of the weaker structures corresponding to modes with small cutoff. These are negligible effects in particular at low
temperature.
In Fig. 6.2 we investigate the single-mode convergence of the most strongly
coupled modes of C60 : Hg 1, Hg 2, Hg 3, Hg 7 (the frequencies and coupling are listed
in Table 3.1), at T = 0 (i.e. for the 0-phonons – ground state – initial excitation). All
the spectra are calculated with the continued-fraction expansion (6.12) on a Lanczos
chain of 300 iterations, and a Lorenzian broadening of 5 meV. For those singlemode systems the mask is simply the max number of phonons, and our truncation
coincides with the standard one. However, is interesting to observe that, in this
medium-coupling regime, except the strongest mode Hg 1, which needs up to 10
phonons to converge completely, all the other modes already with two phonons give
converging results. At this resolution, all these modes show only a one-phonon
satellite at about h̄ωΛj from the main zero-phonon peak. The position of the 1phonon peak is blue-shifted with respect to the original frequency of the mode. It
is also split into two vibronic final structures, only visible for mode Hg 7 in Fig. 6.2
due to the broadening: This is a typical JT-effect, crucial for the interpretation of
the observed C60 photoemission spectrum, as we will show in the next Chapter.
For a many modes system the cutoff must be slightly raised with respect to the
68
CHAPTER 6. DEGENERATE MODES
H1
H2
9
8
Intensity [arb. units]
Intensity [arb. units]
10
6
1
-200 -100
0
100 200 300
Binding Energy [meV]
-200 -100
0
100 200 300
Binding Energy [meV]
H7
Intensity [arb. units]
Intensity [arb. units]
H3
3
3
2
4
4
4
4
3
2
2
1
1
-200 -100
0
100 200 300
Binding Energy [meV]
-200 -100
0
100 200 300
Binding Energy [meV]
Figure 6.2: Convergence of the T = 0 K computed spectra of single degenerate mode, for
the most strongly coupled modes of C 60 , Hg 1, Hg 2, Hg 3, Hg 7 for increasing cutoff of the
basis. All spectra are broadened with Lorenzians of width 5 meV. Except the strongly
coupled Hg 1, (gHg 1 > 3), which needs to include up to 10-phonon states, the other modes
(whose coupling is ' 1) are converged with the inclusion of only the 2-phonons states.
They all show basically one satellite corresponding to the 1-phonon excitation. Note the
splitted JT-vibronic structure of the H g 7 mode, appearing in spite of broadening.
69
6.2. THE JT MODES OF C60 : LANCZOS SPECTRA
mask
{15,8}
{15,6}
{15,4}
Hg1 + Hg5
Intensity [arb.units]
Intensity [arb.units]
Hg1 + Hg3
mask
{15,5}
{15,3}
{15,2}
-100
0
100
Binding Energy [meV]
200
{15,1}
0
Binding Energy [meV]
200
Figure 6.3: For system of many modes the cut off must be lightly relaxed, respect the
values obtained for the single mode, Fig. 6.2, because of the coupling to the strongest
mode Hg 1. The effect is stronger for the strongly coupled mode as H g 3 (on the left),
than for weakly coupled one, as Hg 5 (on the right). All the spectra are broaden with
Lorentians of width 5 meV.
values obtained for the single mode Fig. 6.2. Consider, for example, the 2-mode
system consisting of the most strongly coupled mode Hg 1 plus Hg 3, also significantly
coupled. The left panel of Fig. 6.3 shows the T = 0 spectrum calculated for different
masks. The mask {15,2}, extracted by the single mode investigation (it would be
{10,2} but to add phonons into the strongest coupled mode is almost “free” in
our cutoff scheme, so we stay on the safe side) is far from giving a converging
spectrum, because states with one Hg 3-phonon and more than two Hg 1-phonons
are dropped. Their inclusion, because of the strong coupling of the Hg 1 mode, adds
some broadening to the 1-phonon peak of the Hg 3 mode, as shown in Fig. 6.3. Due
to the low intensity of this peak with respect to the zero-phonon line, relatively few
phonons need to be added and the mask {10,6} already gives a completely converged
spectrum. Moreover for less coupled modes this effect is even less important and
stricter cutoffs can be taken, as shown in Fig. 6.3 (right panel) for the system of
Hg 1 plus the weakly coupled Hg 5.
Finally, in Fig. 6.4, we compare the T = 0 spectra calculated applying both the
standard cutoff vmax = 10 quanta and our strict truncation, with the mask {10,6,3},
chosen on the basis of the 2-mode spectra of Fig. 6.3. With a basis of only 11079
states we get practically the same results as the standard truncation, involving a
basis of 3268760 states. This confirms the validity of our truncation technique, at
least at T = 0.
All these tests prove the validity of our mask-cutoff method, and provide the
ingredients for the calculation of the T = 0 photoemission spectrum for the full
system of all the Hg and Gg degenerates modes of C60 to which we devote the next
70
CHAPTER 6. DEGENERATE MODES
Intensity [arb.units]
H1 + H3 + H5
-200
Standard cutoff
vmax=10
Our cutoff
mask = {10,6,3}
-100
0
100
200
Binding Energy [meV]
300
Figure 6.4: Computed spectrum of a system of the three modes H g 1, Hg 3, Hg 5. We
compare the spectra calculated using the mask {10, 6, 3} with the standard truncation
(vmax = 10), illustrated in Fig. 6.1. The spectra are broadened with Lorenzians of width
5 meV. With a basis of 11079 states we get the same spectra than using the standard,
truncation giving a basis of 3268760 states.
Section. Finally, by including also the contribution of the Ag nondegenerate modes,
as illustrated in Sect, 5.2, we obtain the photoemission spectrum of C60 .
6.3
Degenerate modes contribution to the C60 photoemission spectrum
We come now to the actual calculation of the photoemission spectrum of C60 at
T = 0. In Table 6.1 we list the different masks used to cut off the basis. According
to the analysis of the previous Section, we employ larger cutoff for the strongly
coupled modes, in particular for the modes Hg 1, Hg 2, Hg 7 and Hg 8. The modes
Gg and the modes Hg j with an angle αHj close to 1.5, contribute weakly in the
direction of the D3 minimum and their cutoffs need not be so large.
In Fig. 6.5 we show the result of the calculations corresponding to the masks of
Table 6.1. Already a basis of just about 25000 states, corresponding to mask E gives
all the main structure, although with an excessive spectral weight in the main peak,
71
6.3. PES OF DEGENERATE MODES
Intensity [arb.units]
all degenerate
modes
A
B
C
D
E
-200
0
200 400
Binding Energy [meV]
Figure 6.5: Convergence of the computed photoemission spectrum for the system of
all degenerate Hg and Gg modes. Different spectra correspond to different cutoff masks
listed in Table 6.1. Already with a basis including just about 25000 states (E), the main
structures appear. The spectra evolve to full convergence in the largest calculation (A)
based on ∼ 4.5 million states.
72
CHAPTER 6. DEGENERATE MODES
mask
{H1 H2 H3 H4 H5 H6 H7 H8 G1 G2 G3 G4 G5 G6}
A
B
C
D
E
30
30
20
15
13
8
7
6
5
4
6
6
5
4
3
5
5
5
4
3
4
3
3
3
2
4
3
3
3
2
7
6
5
4
3
7
5
5
4
3
6
6
5
4
3
4
4
3
3
2
5
6
5
4
3
5
5
5
4
3
4
3
3
3
2
4
3
3
3
2
dim
[n.states]
4535498
3073383
1625327
214154
24546
E0
[meV]
-102.5328
-102.0102
-101.9899
-101.6336
-98.8274
Intensity [arb. units]
Table 6.1: We show the mask used to investigate the convergence of the photoemission
spectrum of C60 , and the dimension of the corresponding basis. The results of the calculation are plotted in Fig. 6.5. In the third column we give the ground state energy E 0 ,
whose convergence indicate the convergence of the variational calculation.
all modes
only degenerate
modes
-200
0
200
Binding Energy [meV]
400
Figure 6.6: Total photoemission spectrum of C 60 at T = 0 K, including also the non
degenerate modes, Ag 1 and Ag 2. The effect of the A modes is negligible because of their
weak coupling (Table 3.1).
73
6.4. FINITE-TEMPERATURE LANCZOS
and with a rather poor ground-state energy E0 (JT energy gain) (Table 6.1). As the
calculation is variational with respect to an extension of the basis, the convergence
of the ground state energy E0 is an indicator of the goodness of the calculation.
The values of E0 plus the spectra of Fig 6.5 confirm the validity of the mask-cutoff
technique employed and the good convergence of the spectrum A. We shall take
spectrum A as the starting point of the analyzing of the spectrum in the following.
Finally, we complete the calculation of the T = 0 K photoemission spectrum of
C60 by the inclusion of the contribution of the A modes, as illustrated in Sect. 5.2.
Because of their small coupling their effect is only to slightly broaden globally the
spectrum, as shown in Fig 6.6.
In the next Section we extend the present technique to the calculation of the
photoemission spectrum at finite temperature, investigating the problems to be
faced due to thermal excitation of the initial states.
6.4
Finite-temperature Lanczos
The calculation of the spectrum at finite temperature involves the contribution
of many initial excitations, in accordance with Eq. (4.16). Besides the obvious
problem of requiring a longer computational time, as seen for the nondegenerate
modes, Eq. (5.14), high excitations involves many nonnegligible components on
highly excited states in the finite state basis: a larger basis could be needed to
obtain converging results.
In Fig. 6.7, we investigate the convergence problem for a single 5-phonons initial
excitation, for the strongly coupled modes, Hg 1, Hg 2, Hg 3, Hg 7, as we did in
Fig. 6.2, for the 0-phonons initial state. Up to 20 phonons for Hg 1, and to 8-10 for
Hg 2, Hg 3, Hg 7 are needed to get converging spectra, which requires a much larger
basis than for T = 0 calculation.
The previous contribution of each given excited initial state to the total spectrum
is weighted by the Boltzmann factor e−β Ei , where Ei is the energy of the initial
excitation. Then, for high-frequency modes very few excitation give significant
contribution. Unfortunately, the lowest-frequency mode in C60 , Hg 1, is also the most
strongly coupled one: these two fact end up making finite temperature calculations
rather difficult. The mean number of quanta for an harmonic oscillator of frequency
ω, is given by the Planck’s relation:
n̄ =
1
eβ h̄ω
−1
β=
1
.
kB T
(6.17)
√
We can roughly estimate, that thermal statistics explores states with up to (n̄+ n̄)phonons at a given temperature T . Consequently, to make a converged quantum
74
CHAPTER 6. DEGENERATE MODES
20
17
H2
Intensity [arb. units]
Intensity [arb. units]
H1
10
9
8
15
7
12
6
-200 -100 0 100 200 300
Binding Energy [meV]
-200 -100 0 100 200 300
Binding Energy [meV]
10
9
8
H7
Intensity [arb. units]
Intensity [arb. units]
H3
10
9
8
7
7
6
6
-200 -100 0 100 200 300
Binding Energy [meV]
-200 -100 0 100 200 300
Binding Energy [meV]
Figure 6.7: Convergence of the spectra with starting excitation (1, 1, 1, 1, 1) and M = 2
for increasing cutoff of the oscillators basis. Note the appearance of the 4-phonons antiStokes satellite, besides the main line (5 phonons) and Stokes (6 phonons) satellite for all
relatively weakly coupled modes Hg 2, Hg 3, and Hg 7.
75
6.4. FINITE-TEMPERATURE LANCZOS
30
all
de
ge
ne
ra
te
m
od
es
number of phonons
40
20
10
H1
H7+H8
0
200
400
600
800
1000
T [K]
Figure 6.8: Mean numbers of phonons as a function of temperature.
calculation, it takes a basis of states containing up to about twice that number of
phonons.
In Fig. 6.8 we plot the dependence of n̄ on temperature for the C60 oscillators,
including only the degenerate modes involved in the computation. At T = 800 K,
the temperature
where molecular beam experiments have been carried out, about
√
(25 + 25) ' 30-phonons excitations are be excited, for which a cutoff of 60 quanta,
giving a basis of ∼ 1036 states, would be necessary. This is clearly only a rough,
slightly pessimistic estimate, but it clarifies that there is no hope to compute photoemission spectrum of C60 at 800 K with this technique. We can only get converging
spectra at temperature not higher than about 100 K.
In spite of this, the investigation of the thermal effects in a single-mode spectrum
is perfectly feasible: together with the zero-temperature spectrum of C60 of Fig. 6.5,
allows us to understand all the important features of the measured spectra, as we
will see in Chap. 7.
For single-mode, the spectrum up to 1000 K can be quite easily computed. For
the lowest frequency mode Hg 1, which strongly feels the thermal effects, we have to
consider up to roughly 15-phonons excitations, (see Fig. 6.8), for which a cutoff of
30 phonons is required. The right panel of Fig. 6.9 shows that indeed 30 phonons
(i.e. a basis of 324632 states) gives a well converged thermal spectrum for Hg 1. The
sum of Eq. (4.16) involves an ensemble of more than 15000 states (i.e. 15000 spectra
to compute).
This huge number of diagonalization can be drastically reduced by random sampling. The idea of random sampling [48] is to substitute the sum over the initial
excitations with a sum over a suitable sample granting the correct partition function
CHAPTER 6. DEGENERATE MODES
no sampling
15504 states
Hg1 T=800 K
Ns=300
Ns100
Hg1 T=800 K
Intensity [arb. units]
Intensity [arb. units]
76
cut-off
30
20
10
-200
-100
0
100
Binding Energy [meV]
200
-200
0
Binding Energy [meV]
200
Figure 6.9: Left: sampling convergence. The photoemission spectra at T = 800 K, for
sampling with 100 and 300 states, are compared to the the calculation without sampling
on an ensemble of 15504 states, including all states up to 15-phonons initial excitations.
Right: basis cutoff dependence.
Z=
X
i
e−βEi −→ Z s =
X
A(r)
(6.18)
r∈sample
where A(r) is the correct weight to give to each state Ψr of the sample. For each
possible initial state we generate a random number ζ between 0 and ξ
0<ζ<ξ
(6.19)
where ξ is a number taken in the (0,1] interval in such a way to obtain a sampling
of a given size. A state Ψr is kept in the sample only if e−β Er > ζ. The probability
of keeping a state Ψr , is then
e−β Er
ξ
if ξ > e−β Er
1 if ξ < e−β Er .
Then A(r) is simply the max(e−β Er , ξ) and the photoemission spectrum (4.16) is
given in term of the sample states by
X
1
I(E) = s
(6.20)
max(e−β Er , ξ) Ir (E)
Z
r∈sample
The number of states of the sample depends on the choice of ξ according to
Ns ≈
X e−β Ei
i
ξ
n(Ei )
(6.21)
77
Hg1
T=0
T=800
T=300
Hg 7
Intensity [arb. units]
Intensity [arb. units]
6.4. FINITE-TEMPERATURE LANCZOS
T=0
T=800
-200
-100
0
100
Binding Energy [meV]
200
-200
0
200
Binding Energy [meV]
Figure 6.10: Temperature dependence of the Hg 1 (on the left) and Hg 7 (on the right)
spectra. All spectra have a Lorentzian broadening of 10 meV. Notice the different effect
of temperature. On the lowest frequency modes H1 g a large broadening appears, while
for the highest frequency one Hg 7 only little spectral weight displaces from the main peak
to the satellites.
where n(Ei ) is the number of states of the ensemble with energy Ei . Inverting
Eq. (6.21) we then get a relation to obtain the correct ξ giving the needed sample
dimension Ns . A measure of the goodness of the sample is given by the partition
function error Z s − Z: an error of few per cent indicates a good sample. The left
panel of Fig. 6.9 (Hg 1 at T = 800 K) shows that a sample of few hundred states is
sufficient, with a computational time gain of 50-100 times or more.
Finally, notice the much stronger effect of temperature on the lowest-frequency
modes Hg 1 with respect to the highest frequency one Hg 7 as shown in Fig. 6.10.
For Hg 1, a large broadening appears, giving to the spectrum a Gaussian-like shape,
much different from that at T = 0 K. For Hg 7 instead the spectrum at T = 800 K
remains practically unchanged than at T = 0 K. The only effect of thermal excitations is a minor displacement of spectral weight from the main peak to the satellite.
In particular we can see a weak anti-Stokes structure at about 200 meV below the
zero-phonon line. This is enlightening and suggests where the origin of the large
brodening of the measured spectra is.
In the final Chapter we use the present results to interpret the main features of
the measured spectra.
78
CHAPTER 6. DEGENERATE MODES
Chapter 7
Analysis and discussion
Intensity [arb. units]
Canton et al.
Bruhwiler et al.
8000
7500
Binding energy [meV]
Figure 7.1: The experimental PES by Brühwiler et al [1] and by Canton et al [2].
Photoemission from a closed-shell high-symmetry molecule, involving JT-effect
in the final states, is by necessity accompanied by characteristic vibronic structure
in the measured spectra. In this chapter we employ our theoretical T = 0 photoemission spectrum from the HOMO of C60 , computed in the previous chapter
(Fig. 6.6), to understand and assign single-molecule photoemission spectrum, measured by two separate groups [1, 2] and shown in Fig. 7.1. Unfortunately, both
spectra are measured at T ' 800 K in a molecular beam. They show a strong satellite at 230 meV and a weak shoulder around 400 meV above the main peak, and
79
80
CHAPTER 7. ANALYSIS AND DISCUSSION
Intensity [arb. units]
computed spectrum T=0 K
(A modes included)
measured spectrum T=800 K
230 meV
-400
-200
0
200
400
600
Binding Energy [meV]
Figure 7.2: The computed spectrum at T = 0 compared with the experimental spectrum [1] (T=800 K). The HOMO, prior to JT coupling is taken at 0 energy on this scale
and the experimental spectrum is shifted accordingly. In spite of the large broadening of
the measured spectrum, the strong satellite at 230 meV from the main peak, is the dominant features of both spectra. This makes the T = 0 photoemission spectrum, stripped
of thermal effects, the correct starting point for the understanding of these typical JT
features.
a characteristic large asymmetric broadening. This 230 meV separation is some
20% larger than the highest characteristic frequency of C60 , and thus cannot be
understood in terms of a simple non-JT electron-phonon model (see Sect. 5.4 and
Ref. [1]).
The gigantic vibrational Hilbert space size called for at this high temperature
prevents us from getting converging spectra with the technique described in the
previous chapter. However, the T = 0 spectrum already shows the main satellite
at the right position, plus several other structures hidden by thermal effects in the
observed spectrum. Analyzing in detail the different contributions to the computed
spectrum, and investigating thermal effects on the more accessible single-mode systems, we account satisfactorily for the experimental lineshape.
7.1
T = 0 analysis
In Fig. 7.2, we compare the T = 0 computed spectrum, with the measured one
(Ref. [1]). In spite of the impossibility of an all-modes finite-temperature calcu-
7.1. T = 0 ANALYSIS
81
analysis C60
Intensity [arb. units]
all modes
only Hg modes
Hg1+Hg2+Hg3+Hg7+Hg8
Hg1+Hg2+Hg7+Hg8
Hg1+Hg7+Hg8
0
200
400
Binding Energy [meV]
600
Figure 7.3: Analysis of the different contribution to the T = 0 computed spectrum. The
investigation shows that the main features of the complete spectrum are accounted for by
the two high-frequency modes, Hg 7 and Hg 8, responsible of the strong 230 meV structure
and the weak 400 meV shoulder in the measured spectra, and by H g 1, responsible for the
∼ 30 meV close to the main peak.
lation discussed in Sect. 6.4, we see that already the T = 0 computed spectrum
shows an important structure at 230 meV in correspondence to the shoulder of the
experiment, plus several extra structures apparently hidden by thermal effects in
the experiment.
We analyze now the different contributions, in order to extract a more general
qualitative understanding of the spectrum. As a first step, we remove sequentially
the contributions of the weakly coupled modes, and of the modes which do not
participate directly to the 230 meV structure. In Fig. 7.3 we show the sequence of
this analysis. We ignore the reduction in JT energy gain, which takes contributions
from all modes, by shifting the ground-state energy of all the spectra to zero energy. The investigation shows that the main features of the complete spectrum are
accounted for by the two high-frequency modes, Hg 7 and Hg 8, responsible for the
82
CHAPTER 7. ANALYSIS AND DISCUSSION
Intensity [arb. units]
analysis
Hg1 Hg7 Hg8
Hg7+Hg8
Hg1
Hg1+Hg7+Hg8
0
200
400
Binding Energy [meV]
Figure 7.4: Analysis of the specific contributions of the H g 1 and of Hg 7 plus Hg 8. When
only the Hg 1 mode is considered all satellite structures above 200 meV disappear demonstrating that the 230 meV spectral structure in C 60 is to be associated to vibrons connected
with the high-frequency modes Hg 7 and Hg 8. Note the blue-shift of the satellites in the
Hg 7 + Hg 8 spectrum.
strong 230 meV structure and the weak 400 meV shoulder in the measured spectra,
and by Hg 1, responsible for the ∼ 30 meV sequence close to the main peak. The
other modes contribute weakly in the intermediate range of energy 50 − 150 meV
(mainly Hg 2 and Hg 3). The exclusion of all the Ag , Gg , and of Hg 4, Hg 5, Hg 6
modes leaves the spectrum almost unaffected, because of their very weak couplings.
In Fig. 7.4 we show the specific contributions of the Hg 1 and of Hg 7 plus Hg 8:
Hg 7 and Hg 8 affect the same region of the spectrum, because of their comparable
frequencies and couplings. When only the Hg 1 mode is considered, all satellite structures above 200 meV disappear demonstrating that the 230 meV spectral structure
in C60 is to be associated to vibrons connected with the high-frequency modes Hg 7
and Hg 8. The spectrum computed including these two modes alone, shows a narrow
satellite above 200 meV, This main satellite appears significantly (10%) blue-shifted
with respect to h̄ωHg 8 . This increase in vibron excitation energy above the purely
vibrational h̄ω for the 1-phonon excitation of a trivial displaced oscillator is in fact
a characteristic signature of dynamic JT effect, and can be understood on the basis
83
α=0.5
average
(8)
t1u
& Hg
0.5
gu
au
Hg
Hg 7
Hg7+Hg8
hu (GS)
0
0
H g8
Intensity [arb. units]
1
(7)
(Energy-EGS) / hω
t2u
hω8
hu
hω7
7.2. 1-PHONON VIBRONIC MULTIPLET INTERPRETATION
2
4
Jahn-Teller coupling g
6
175
200
225
250
Binding Energy [meV]
Figure 7.5: Left panel: Vibronic excitation energies for a single H g mode coupled to the
hu level, as a function of coupling g. The value α = 0.5 is characteristic of the H g 7 and
Hg 8 modes. The average refers to the multiplet originating from the 1-phonon states. h u
states, the only visible states in T = 0 PES, are highlighted as thick solid lines. Right
panel: detail of the Hg 7, Hg 8 1-phonon structures (Lorenzian = 1 meV). Note the
blue-shift of the average of the hu vibrons, and their cooperation to the shifted peak in
Hg 7 + Hg 8.
of symmetry. In the next Section we investigate quantitatively this effect for T = 0,
where we have a complete understanding of the spectral features, leaving to the last
Section the extension to the finite temperature.
7.2
1-phonon vibronic multiplet interpretation
The blue-shifted peak of the Hg 7 + Hg 8 spectrum of Fig. 7.4, is readily explained
on the basis of symmetry. We will argue in the next Section that the conclusions
we arrive to here extend to finite temperature T = 800 K.
At T = 0 the only contribution to the spectrum (4.16) comes from the ground
state Ψi=0 , and the bare excitation c† |Ψ0 i has the orbital hu symmetry of the
HOMO. This state has nonzero overlap strictly only with final states of the same
symmetry hu . This single symmetry channel thus represents the only contributor
to the zero-temperature photoemission spectrum.
In the left panel of Fig. 7.5 we depict the vibron excitation energy with special
attention to the hu states, for the simplest case of a single Hg mode, as a function
of the coupling strength g. The average of the multiplet of states derived from
84
CHAPTER 7. ANALYSIS AND DISCUSSION
hu
hu
au
α=0.5
t1u
t2u
gu
0.5
0.5
average
gu
au
Hg
α=0.1
hu (GS)
0
0
hu (GS)
0
2
4
6
0
2
Jahn-Teller coupling g
au
t2u
(Energy-EGS) / hω
ge
hu
t1u
gu
au
2
6
aver
age
gu
au
α=1.5
4
hu
t2u
t1u
hu (GS)
Jahn-Teller coupling g
6
gu
hu
hu
gu
avera
α=1
4
Jahn-Teller coupling g
(Energy-EGS) / hω
hu
0
t1u
(7)
au
1
(8)
average
& Hg
1
(Energy-EGS) / hω
(Energy-EGS) / hω
t2u
0
hu (GS)
2
4
6
Jahn-Teller coupling g
Figure 7.6: α dependence of 1-vibron multiplet pattern. The h u states are blue-shifted
for any value of α. However, α is crucial in determining the precise splitting pattern of
the 1-vibron multiplet.
the 1-vibration manifold is, to second order in the electron-phonon coupling g,
independent of g, because the JT coupling is a traceless perturbation within that
manifold. Within this multiplet (made of Hg × hu = au + t1u + t2u + 2gu + 2hu ), the
hu vibronic states are the sole that have another hu state (the GS itself) lower in
energy and “pushing” them upward. As a result, for small but increasing coupling,
the hu states shift in average necessarily upward in energy above h̄ωHg . The amount
of this shift reflects the strength of the coupling.
In the right panel of Fig. 7.5 we show the 1-phonon vibronic structure of the
two high-frequency modes Hg 7, Hg 8 (at h̄ω = 181 and 197 meV) and of their
simultaneous action Hg 7 + Hg 8. The couplings for these two modes alone accounts
for a 10%, or roughly 20 meV shift of the important Hg 7 + Hg 8 vibron satellite
above h̄ω. This is about half of the total observed and calculated shift of ∼ 40 meV
7.3. THERMAL EFFECTS
85
of the 230 meV satellite relative to h̄ω = 181 ÷ 197 meV. The crucial observation
here is the asymmetry of the spectral weight distribution between the pair of hu
vibronic states, for both the two modes Hg 7, Hg 8. This manifests itself as a blueshifted main satellite also above the multiplet average. Moreover, the cooperation
of the two modes amplifies this effect in the Hg 7 + Hg 8 photoemission spectrum
(Fig. 7.5). The remaining 10% blue shift is then a collective effect of all other
modes simultaneously interacting with the HOMO, and pushing their respective
1-vibration satellite upward. All goes as if effectively the coupling of the high
frequency Hg 7, Hg 8 modes were larger, i.e. effectively moving to the right of the
vertical dashed line in the left panel of Fig. 7.5.
So far, we referred just to the Hg 7, Hg 8 coupling strength as the main parameter.
However, as discussed in Sect.3.4, the H ⊗ h problem is characterized by a second
parameter, namely the value of the mixing angle α [3, 38]. As it turns out, we find
that in C60 this parameter is far from irrelevant and in fact very important. In
static JT, it determines the symmetry of the distortion (D5 vs D3 ). In dynamic
JT, it also determines the ground state symmetry, namely hu for α ≈ 0, and au
for α ≈ π/2 [38]. In PE spectroscopy, α is crucial in determining the precise
splitting pattern of the 1-vibron multiplet, even though the general feature that on
average the hu states shift upward holds for any value of α, as shown in√Fig. 7.6
for α = 0.1, 0.5, 1, and 1.5. En passant, note that for D5 minima (α < 3/ 5) only
an hu and an au state move to low energy at strong coupling: these
√ become the
tunneling states among the 6 D5 wells. For the 10 D3 wells (α > 3/ 5), an extra gu
state also moves to low energy at strong coupling: that one completes the tunneling
picture. Note, however, that the tunneling regime (g ≥ 4) is not relevant for the
intermediate coupling of C60 , contrary to the assignment of Ref. [2, 50].
7.3
Thermal effects
Will the considerations of the previous Section on the 1-phonon states remain valid
at finite-temperature? We claim that this is the case. For the two high-frequency
modes Hg 7 and Hg 8, at T = 800 K, the thermal energy kB T is only about one third
of h̄ω: the main contribution to the spectrum still comes from the ground-state.
Indeed their spectrum lineshape is practically unchanged upon turning temperature
up to 800 K, as shown in the left panel of Fig. 7.7. The T = 0 conclusions of the
previous Sections, still hold at T = 800 K: the main peak at 230 meV comes from
the splitting of the 1-phonon excitation of Hg 7 plus Hg 8. Thermal effects only
affect significantly the low frequency modes, especially Hg 1. for which many initial
excitations have to be included. The broadening of its T = 800 K spectrum accounts
for most of the observed broadening of the band, as shown in Fig. 7.7 (right), This
broadening is responsible for hiding the minor structures of the spectrum. Many
86
CHAPTER 7. ANALYSIS AND DISCUSSION
Figura 3
Hg7+Hg8
T=800 K
solo il modo Hg a 32 meV
intensity (arb. units)
Intensity [arb. units]
spettro calcolato (T=800K)
spettro sperimentale (T=800K)
T=0 K
0
200
-400
-200
Binding Energy [meV]
0
200
400
600
energy (meV)
Figure 7.7: Left panel: T -dependence of the Hg 7 plus Hg 8 spectrum. At T = 800 K
the main contribution to the photoemission spectra still comes from the ground state:
no substantial change in the spectrum lineshape appears. Right panel: T = 800 K H g 1
spectrum (Lorenzian broadening = 10 meV compared to the width of the experimental
band).
modes effects destroy the residual coherence seen in the Hg 1-only spectrum.
7.4
Conclusions and outlook
The results of the present calculations confirm that the linear JT model (3.6) based
on the DFT-LDA parameters of Ref. [3] describes well the electron-phonon dynamics
of C+
60 . The difficulties of the present technique to address high temperature can
be overcome by a combined use of random sampling (Sect 6.4) and a variablebasis method, where the quantum basis adapts itself to each sampled initial state,
including only those states most strongly coupled to it. The implementation of such
technique is currently being pursued, with good preliminary results, not included
in this work.
A quantitative understanding of the photoemission spectrum from a degenerate molecular shell, with its characteristic vibronic features, is mandatory if the
all-important electron-vibration coupling parameters are to be extracted from this
kind of experiment, often the only reliable source of pertinent information available
on the molecular ion. The method employed here has wide potential application either based, as in this work, on ab initio parameters, or, for simple enough molecules,
used to determine electron-phonon parameters by means of suitable fitting of experimental data.
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